5.01a Permutations and combinations: evaluate probabilities

1 The 26 members of the local sports club include Mr and Mrs Khan and their son Abad. The club is holding a party to celebrate Abad's birthday, but there is only room for 20 people to attend. In how many ways can the 20 people be chosen from the 26 members of the club, given that Mr and Mrs Khan and Abad must be included?

5 A security code consists of 2 letters followed by a 4-digit number. The letters are chosen from \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) and the digits are chosen from \(\{ 1,2,3,4,5,6,7 \}\). No letter or digit may appear more than once. An example of a code is BE 3216 .

1. How many different codes can be formed?
2. Find the number of different codes that include the letter A or the digit 5 or both.
   A security code is formed at random.
3. Find the probability that the code is DE followed by a number between 4500 and 5000 .

6 A Social Club has 15 members, of whom 8 are men and 7 are women. The committee of the club consists of 5 of its members.

1. Find the number of different ways in which the committee can be formed from the 15 members if it must include more men than women.
   The 15 members are having their photograph taken. They stand in three rows, with 3 people in the front row, 5 people in the middle row and 7 people in the back row.
2. In how many different ways can the 15 members of the club be divided into a group of 3, a group of 5 and a group of 7 ?
   In one photograph Abel, Betty, Cally, Doug, Eve, Freya and Gino are the 7 members in the back row.
3. In how many different ways can these 7 members be arranged so that Abel and Betty are next to each other and Freya and Gino are not next to each other?
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

6

1. Find the number of different arrangements of the 9 letters in the word ACTIVATED.
2. Find the number of different arrangements of the 9 letters in the word ACTIVATED in which there are at least 5 letters between the two As.
   Five letters are selected at random from the 9 letters in the word ACTIVATED.
3. Find the probability that the selection does not contain more Ts than As.

6 \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-12_291_809_255_667} In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, \(X\) and \(Y\). Rajid, Sue and Tan are three of these friends.

1. The eight friends will be divided into two groups of 4, one group for table \(X\) and one group for table \(Y\). Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.
   When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table \(X\) on the same side as each other. Tan decides that he does not mind at which table he sits.
2. Find the number of different seating arrangements for the 8 friends.
   As they leave the restaurant, the 8 friends stand in a line for a photograph.
3. Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)
2. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together.
   Four letters are selected at random from the 9 letters in the word ANDROMEDA.
3. Find the probability that this selection contains at least one D and exactly one A .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.

1. At the beginning of the party, the 12 people will stand in a line for a photograph.
   1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
   2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as each other.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
   Four letters are selected at random from the 9 letters in the word INTELLECT.
   [0pt]
3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}

2

1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}

6

1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-10_2715_35_110_2012}
3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
   The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
4. Find the probability that both Ps are in one group and both Ss are in the other group.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 A committee of 5 people is to be chosen from 6 men and 4 women. In how many ways can this be done

1. if there must be 3 men and 2 women on the committee,
2. if there must be more men than women on the committee,
3. if there must be 3 men and 2 women, and one particular woman refuses to be on the committee with one particular man?

2

1. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
2. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.

4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?

5

1. The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
   Melon
   or
   Soup
   or
   Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
   Chicken
   or
   Steak
   or
   Lamb Cutlets
   or
   Vegetable Curry
   or
   Fish \end{displayquote} \begin{displayquote} Dessert Course
   Cheesecake
   or
   Ice Cream
   or
   Apple Pie
   All the main courses are served with salad and either
   new potatoes or french fries.
   1. How many different three-course meals are there?
   2. How many different choices are there if customers may choose only two of the three courses?
2. In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}

7

1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
   1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
   2. if exactly one forward player must be included, together with any two others.
2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
   1. if there are no restrictions on the order of the letters,
   2. if the two letters D come first and the two letters L come last.

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?

5

1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
   1. if there are no restrictions,
   2. if the Rs must all be together.
   3. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?

3 Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.

1. How many different arrangements of all 11 CDs on a shelf are there if the jazz CDs are all next to each other?
2. Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different possible selections can be made?

4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.

1. In how many different ways can the group be chosen?
2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?

6

1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.

1 A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are selected at random. Find the probability that exactly 3 of them are red.

4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

1. the first and last cans in the row are the same type of drink,
2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.

4 A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers and 2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least 1 wicketkeeper.

1. Find the number of different ways in which the team can be chosen. Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries and 2 identical notebooks.
2. Find the number of different arrangements of the presents if they are all displayed in a row.
3. 10 of these 11 presents are chosen and arranged in a row. Find the number of different arrangements that are possible.

4

1. Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in a line.
2. The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability that the 3 vowels (A, E, I) are not all next to each other.
3. Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss.