Permutations & Arrangements

169 questions · 29 question types identified

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Linear programming and optimization

Questions involving graphical solutions to linear programming problems, constraint satisfaction, or finding optimal solutions through graphical methods or simplex-type approaches.

18 Easy -1.8
10.7% of questions
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Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
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Easiest question Easy -2.5 »
Describe one advantage and one disadvantage of
  1. quota sampling, [2]
  2. simple random sampling. [2]
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Hardest question Challenging +1.2 »
Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
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Digit arrangements forming numbers

A question is this type if and only if it involves arranging digits to form numbers with constraints (e.g., 'even numbers between 3000 and 5000').

15 Moderate -0.4
8.9% of questions
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3 Numbers are formed using some or all of the digits 4, 5, 6, 7 with no digit being used more than once.
  1. Show that, using exactly 3 of the digits, there are 12 different odd numbers that can be formed.
  2. Find how many odd numbers altogether can be formed.
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Easiest question Easy -1.2 »
  1. In lines 8 to 10, the article says "Some banks do not allow numbers that begin with zero, numbers in which the digits are all the same (such as 5555) or numbers in which the digits are consecutive (such as 2345 or 8765)." How many different 4-digit PINs can be made when all these rules are applied? [3]
  2. At the time of writing, the world population is \(6.7 \times 10^9\) people. Assuming that, on average, each person has one card with a 4-digit PIN (subject to the rules in part (i) of this question), estimate the average number of people holding cards with any given PIN. Give your answer to an appropriate degree of accuracy. [2]
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Hardest question Challenging +1.2 »
    1. The polynomial \(\mathrm { F } ( x )\) is a quartic such that
$$\mathrm { F } ( x ) = p x ^ { 4 } + q x ^ { 3 } + 2 x ^ { 2 } + r x + s$$ where \(p , q , r\) and \(s\) are distinct constants.
Determine the number of possible quartics given that
  1. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,1,3,5 \}\)
  2. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,0,1,3,5 \}\) (ii) A 3-digit positive integer \(N = a b c\) has the following properties
    • \(N\) is divisible by 11
    • the sum of the digits of \(N\) is even
    • \(N \equiv 8 \bmod 9\)
    • Use the first two properties to show that
    $$a - b + c = 0$$
  3. Hence determine all possible integers \(N\), showing all your working and reasoning.
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Arrangements with positional constraints

A question is this type if and only if it requires arranging letters/objects with specific items in fixed positions (e.g., 'E at beginning and end', 'R in central position').

13 Moderate -0.1
7.7% of questions
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3 The six digits 4, 5, 6, 7, 7, 7 can be arranged to give many different 6-digit numbers.
  1. How many different 6-digit numbers can be made?
  2. How many of these 6-digit numbers start with an odd digit and end with an odd digit?
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Easiest question Moderate -0.8 »
6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if
  1. there are no restrictions,
  2. the first letter is R and the last letter is G ,
  3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
  4. Find the number of selections which contain no Es and exactly 1 R .
  5. Find the number of selections which contain no Es.
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Hardest question Standard +0.8 »
6
  1. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
  2. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
  3. Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.
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Arrangements with adjacency requirements

A question is this type if and only if it requires certain items to be together (adjacent) in the arrangement.

13 Moderate -0.3
7.7% of questions
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Find the number of different ways that 6 boys and 4 girls can stand in a line if
  1. all 6 boys stand next to each other, [3]
  2. no girl stands next to another girl. [3]
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Easiest question Easy -1.3 »
1
  1. The letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E are arranged in a straight line.
    1. How many different arrangements are possible?
    2. In how many of these arrangements are the letters A and B next to each other?
    3. From the letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , two different letters are selected at random. Find the probability that these two letters are A and B .
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Hardest question Standard +0.3 »
7
  1. Find the number of different arrangements of the 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.
  2. The 9 letters in the word ALLIGATOR are arranged in a random order. Find the probability that the two Ls are together and there are exactly 6 letters between the two As.
  3. Find the number of different selections of 5 letters from the 9 letters in the word ALLIGATOR which contain at least one A and at most one L.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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Arrangements with grouped categories

A question is this type if and only if it requires all items of certain categories to be grouped together (e.g., 'all vowels together and all consonants together').

10 Moderate -0.2
5.9% of questions
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4
  1. Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together.
  2. Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy.
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Easiest question Moderate -0.8 »
6
  1. Find the number of ways in which all 9 letters of the word AUSTRALIA can be arranged in each of the following cases.
    1. All the vowels (A, I, U are vowels) are together.
    2. The letter T is in the central position and each end position is occupied by one of the other consonants (R, S, L).
  2. Donna has 2 necklaces, 8 rings and 4 bracelets, all different. She chooses 4 pieces of jewellery. How many possible selections can she make if she chooses at least 1 necklace and at least 1 bracelet?
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Hardest question Standard +0.8 »
7
  1. How many different arrangements are there of the 10 letters in the word REGENERATE?
  2. How many different arrangements are there of the 10 letters in the word REGENERATE in which the 4 Es are together and the 2 Rs have exactly 3 letters in between them?
  3. Find the probability that a randomly chosen arrangement of the 10 letters in the word REGENERATE is one in which the consonants ( \(\mathrm { G } , \mathrm { N } , \mathrm { R } , \mathrm { R } , \mathrm { T }\) ) and vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { E } , \mathrm { E }\) ) alternate, so that no two consonants are next to each other and no two vowels are next to each other. [5]
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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Arrangements with identical objects

A question is this type if and only if it involves arranging multiple identical objects of different types (e.g., 'identical red, blue, and yellow mugs').

8 Moderate -0.3
4.7% of questions
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2 Twelve coins are tossed and placed in a line. Each coin can show either a head or a tail.
  1. Find the number of different arrangements of heads and tails which can be obtained.
  2. Find the number of different arrangements which contain 7 heads and 5 tails.
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Correct ordering probability

Questions asking for the probability that items appear in a specific correct order or nearly correct order among all possible arrangements.

8 Standard +0.1
4.7% of questions
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I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.
  1. If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left. [2]
  2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. [3]
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Arrangements with alternating patterns

A question is this type if and only if it requires alternating types of items (e.g., consonant-vowel-consonant pattern, or no two of same type adjacent).

7 Standard +0.4
4.1% of questions
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1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.
  1. Find the number of different arrangements using all nine letters.
  2. How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?
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Selection with type constraints

A question is this type if and only if it asks for the number of ways to select items with constraints on how many of each type (e.g., 'at least one M and exactly one E').

6 Easy -1.0
3.6% of questions
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  1. State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey. [2]
  2. State one advantage and one disadvantage of quota sampling. [2]
(Total 4 marks)
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Basic arrangements with repeated letters

Questions asking for the number of arrangements of letters from a word containing repeated letters, with no additional constraints beyond arranging all or some of the letters.

6 Moderate -0.6
3.6% of questions
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1
  1. How many different arrangements are there of the 11 letters in the word MISSISSIPPI?
  2. Two letters are chosen at random from the 11 letters in the word MISSISSIPPI. Find the probability that these two letters are the same.
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Items NOT together (general separation)

Questions requiring certain items to NOT be together or adjacent, solved by subtracting 'together' arrangements from total arrangements or using gap methods.

6 Moderate -0.3
3.6% of questions
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1 A group consists of 5 men and 2 women. Find the number of different ways that the group can stand in a line if the women are not next to each other.
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Specific items separated

Questions asking for the probability that specific items are NOT together or are separated (e.g., no two men next to each other, vowels not all together).

6 Standard +0.7
3.6% of questions
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The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other. [3]
  2. Find the probability that no two men are next to one another. [4]
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Sorting and searching algorithms

Questions about sorting algorithms (bubble sort, quick sort) or searching algorithms (binary search) applied to lists of numbers or names, including analysis of algorithm steps and iterations.

6 Moderate -1.0
3.6% of questions
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2
2
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\hline &
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\hline
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People arrangements in groups/rows

A question is this type if and only if it involves arranging people into multiple groups, rows, or teams with specific composition requirements.

5 Standard +0.2
3.0% of questions
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6
  1. Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
  2. Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.
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Code/password formation

A question is this type if and only if it involves forming codes or passwords from letters and/or digits with repetition constraints.

5 Moderate -0.4
3.0% of questions
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2 Codes of three letters are made up using only the letters A, C, T, G. Find how many different codes are possible
  1. if all three letters used must be different,
  2. if letters may be repeated.
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Specific items together

Questions asking for the probability that specific named items or groups are adjacent/next to each other in an arrangement.

5 Moderate -0.2
3.0% of questions
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4 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other.
  2. Find the probability that no two men are next to one another.
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People arrangements in lines

A question is this type if and only if it involves arranging people in a line with constraints about who stands next to whom or in what positions.

4 Moderate -0.4
2.4% of questions
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6 Six men and three women are standing in a supermarket queue.
  1. How many possible arrangements are there if there are no restrictions on order?
  2. How many possible arrangements are there if no two of the women are standing next to each other?
  3. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?
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Exactly N letters between items

Questions requiring exactly a specific number of letters between two particular items, solved by fixing positions with the required gap.

4 Standard +0.7
2.4% of questions
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4
  1. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
  2. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
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Seating arrangements with constraints

A question is this type if and only if it involves arranging people in specific seating configurations (e.g., minibus seats, rows) with positional constraints.

3 Moderate -0.4
1.8% of questions
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4 \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733} The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.
  1. How many possible seating arrangements are there for the 11 passengers?
  2. How many possible seating arrangements are there if 5 particular people sit in the back row? Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.
  3. In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?
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Arrangements with couples/pairs

A question is this type if and only if it involves arranging people where some are couples/pairs with constraints about partners standing together or apart.

3 Moderate -0.4
1.8% of questions
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3 Mr and Mrs Keene and their 5 children all go to watch a football match, together with their friends Mr and Mrs Uzuma and their 2 children. Find the number of ways in which all 11 people can line up at the entrance in each of the following cases.
  1. Mr Keene stands at one end of the line and Mr Uzuma stands at the other end.
  2. The 5 Keene children all stand together and the Uzuma children both stand together.
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Combinatorial selection with category constraints

Questions asking for the number of ways to select a specified number of items from multiple distinct categories where each category has a minimum or maximum constraint (e.g., select 2 pop CDs, 2 jazz CDs, and 1 classical CD).

3 Moderate -0.9
1.8% of questions
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Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
  1. no student may win more than 1 prize, [2]
  2. no student may win all 3 prizes. [2]
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At least/at most N letters between items

Questions requiring at least or at most a specific number of letters between items, typically solved by casework or complementary counting.

3 Standard +0.8
1.8% of questions
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The letters of the word CHAFFINCH are written on cards.
  1. In how many ways can the letters be rearranged with no restrictions. [1]
  2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]
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Alternating pattern probability

Questions asking for the probability of an alternating arrangement pattern between two types of items.

3 Challenging +1.1
1.8% of questions
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  1. Four men and four women stand in a random order in a straight line. Determine the probability that no one is standing next to a person of the same gender. [3]
  2. \(x\) men, including Mr Adam, and \(x\) women, including Mrs Adam, are arranged at random in a straight line. Show that the probability that Mr Adam is standing next to Mrs Adam is \(\frac{1}{x}\). [3]
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Graph theory problems

Questions about graph properties, Eulerian paths, spanning trees, network algorithms (Kruskal's, etc.), or graph construction with specified vertex orders and arcs. These are pure graph theory questions not involving assignment or matching of people to tasks.

3 Moderate -0.3
1.8% of questions
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9. The continuous random variable \(X\) has a uniform distribution on the interval \([ - \pi , \pi ]\).
The random variable \(Y\) is defined by \(Y = \sin X\). Determine the cumulative distribution function of \(Y\). END OF TEST
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Conditional probability in arrangements

A question is this type if and only if it asks for a conditional probability given information about an arrangement (e.g., 'probability given that certain items are together').

2 Standard +0.6
1.2% of questions
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5 The 8 letters in the word RESERVED are arranged in a random order.
  1. Find the probability that the arrangement has V as the first letter and E as the last letter.
  2. Find the probability that the arrangement has both Rs together given that all three Es are together.
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Multiple separation conditions combined

Questions with multiple simultaneous separation requirements (e.g., some items together AND other items not together, or fixed ends with separation constraints).

2 Standard +0.3
1.2% of questions
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4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.
  1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
  2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.
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Optimization assignment problems

Questions where people/teams must be assigned to tasks/jobs/locations with numerical costs or scores in a table, requiring finding the optimal (minimum cost or maximum score) assignment using the Hungarian algorithm or similar methods.

1 Moderate -0.8
0.6% of questions
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5 Viola and Orsino are arguing about which striker to include in their fantasy football team. Viola prefers Rocinate, who creates lots of goal chances, but is less good at converting them into goals. Orsino prefers Quince, who is not so good at creating goal chances, but who is better at converting them into goals. The information for Rocinate and Quince is shown in the tables.
\multirow{2}{*}{}Number of chances created per match
RocinateQuince
Number67895678
Probability\(\frac { 1 } { 20 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)
Probability of converting a chance into a goal
RocinateQuince
0.10.12
  1. Give an efficient rule for using 2-digit random numbers to simulate the number of chances created by Rocinate in a match.
  2. Give a rule for using 2-digit random numbers to simulate the conversion of chances into goals by Rocinate.
  3. Your Printed Answer Book shows the result of simulating the number of goals scored by Rocinate in nine matches. Use the random numbers given to complete the tenth simulation, showing which of your simulated chances are converted into goals.
  4. Give an efficient rule for using 2-digit random numbers to simulate the number of chances created by Quince in a match.
  5. Your Printed Answer Book shows the result of simulating the number of goals scored by Quince in nine matches. Use the random numbers given to complete the tenth simulation, showing which of your simulated chances are converted into goals.
  6. Which striker, if any, is favoured by the simulation? Justify your answer.
  7. How could the reliability of the simulation be improved?
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Arrangements with divisibility constraints

Questions asking for arrangements of digits (not letters) from a number with repeated digits, where the resulting number must satisfy a divisibility or parity condition (e.g., even numbers).

1 Moderate -0.3
0.6% of questions
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7
  1. Find how many different numbers can be made by arranging all nine digits of the number 223677888 if
    1. there are no restrictions,
    2. the number made is an even number.
  2. Sandra wishes to buy some applications (apps) for her smartphone but she only has enough money for 5 apps in total. There are 3 train apps, 6 social network apps and 14 games apps available. Sandra wants to have at least 1 of each type of app. Find the number of different possible selections of 5 apps that Sandra can choose.
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Bipartite matching problems

Questions involving bipartite graphs or tables where people/workers must be matched to tasks/jobs with constraints on who can do what, typically requiring determination of whether a complete matching exists or finding all possible matchings through systematic enumeration. No optimization of costs/scores involved.

0
0.0% of questions
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1 Six people, \(A , B , C , D , E\) and \(F\), are to be allocated to six tasks, 1, 2, 3, 4, 5 and 6. The following bipartite graph shows the tasks that each of the people is able to undertake. \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-02_1003_547_740_737}
  1. Represent this information in an adjacency matrix.
  2. Initially, \(B\) is assigned to task 4, \(C\) to task 3, \(D\) to task 1, \(E\) to task 5 and \(F\) to task 6. By using an algorithm from this initial matching, find a complete matching.
    (3 marks)
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