5.01a Permutations and combinations: evaluate probabilities

7 Find the number of ways the 9 letters of the word SEVENTEEN can be arranged in each of the following cases.

1. One of the letter Es is in the centre with 4 letters on either side.
2. No E is next to another E.
   5 letters are chosen from the 9 letters of the word SEVENTEEN.
3. Find the number of possible selections which contain exactly 2 Es and exactly 2 Ns.
4. Find the number of possible selections which contain at least 2 Es.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 Freddie has 6 toy cars and 3 toy buses, all different. He chooses 4 toys to take on holiday with him.

1. In how many different ways can Freddie choose 4 toys?
2. How many of these choices will include both his favourite car and his favourite bus?
   Freddie arranges these 9 toys in a line.
3. Find the number of possible arrangements if the buses are all next to each other.
4. Find the number of possible arrangements if there is a car at each end of the line and no buses are 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. A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.
2. Find the number of different 7-digit numbers which can be formed from the seven digits 2, 2, 3, 7, 7, 7, 8 in each of the following cases.
   1. The odd digits are together and the even digits are together.
   2. The 2 s are not together.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

4

1. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee.
2. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together.

6 Hannah chooses 5 singers from 15 applicants to appear in a concert. She lists the 5 singers in the order in which they will perform.

1. How many different lists can Hannah make? Of the 15 applicants, 10 are female and 5 are male.
2. Find the number of lists in which the first performer is male, the second is female, the third is male, the fourth is female and the fifth is male. Hannah's friend Ami would like the group of 5 performers to include more males than females. The order in which they perform is no longer relevant.
3. Find the number of different selections of 5 performers with more males than females.
4. Two of the applicants are Mr and Mrs Blake. Find the number of different selections that include Mr and Mrs Blake and also fulfil Ami's requirement.

2 A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons are selected at random without replacement. Find the probability that exactly 3 of them are green.

5

1. A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes.
2. Another plate holds 7 cup cakes, each with a different colour icing, and 4 brownies, each of a different size. Find the number of different ways these 11 cakes can be arranged in a row if no brownie is next to another brownie.
3. A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together.

7 Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.

1. There are no restrictions.
2. The As occupy the 1st, 5th and 9th positions.
3. There is exactly one letter between the Ms.
   Three letters are selected from the 9 letters of the word CAMERAMAN.
4. Find the number of different selections if the three letters include exactly one M and exactly one A.
5. Find the number of different selections if the three letters include at least one M.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 In a certain hotel, the lock on the door to each room can be opened by inserting a key card. The key card can be inserted only one way round. The card has a pattern of holes punched in it. The card has 4 columns, and each column can have either 1 hole, 2 holes, 3 holes or 4 holes punched in it. Each column has 8 different positions for the holes. The diagram illustrates one particular key card with 3 holes punched in the first column, 3 in the second, 1 in the third and 2 in the fourth. \includegraphics[max width=\textwidth, alt={}, center]{2bcbd4d3-0d41-48fa-8f70-192b158c0bbe-2_410_214_1811_968}

1. Show that the number of different ways in which a column could have exactly 2 holes is 28 .
2. Find how many different patterns of holes can be punched in a column.
3. How many different possible key cards are there?

6

1. A collection of 18 books contains one Harry Potter book. Linda is going to choose 6 of these books to take on holiday.
   1. In how many ways can she choose 6 books?
   2. How many of these choices will include the Harry Potter book?
2. In how many ways can 5 boys and 3 girls stand in a straight line
   1. if there are no restrictions,
   2. if the boys stand next to each other?

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?

6 A box contains five balls numbered \(1,2,3,4,5\). Three balls are drawn randomly at the same time from the box.

1. By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number. The random variable \(L\) denotes the largest of the three numbers drawn.
2. Find the probability that \(L\) is 4 .
3. Draw up a table to show the probability distribution of \(L\).
4. Calculate the expectation and variance of \(L\).

3 A staff car park at a school has 13 parking spaces in a row. There are 9 cars to be parked.

1. How many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
2. How many different arrangements are there if the 4 empty spaces are next to each other?
3. If the parking is random, find the probability that there will not be 4 empty spaces next to each other.

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?

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?

4 A builder is planning to build 12 houses along one side of a road. He will build 2 houses in style \(A\), 2 houses in style \(B , 3\) houses in style \(C , 4\) houses in style \(D\) and 1 house in style \(E\).

1. Find the number of possible arrangements of these 12 houses.

2. Road
   \(\square \square \square \square \square \square \square \square \square\)	\(\square \square \square\)

   The 12 houses will be in two groups of 6 (see diagram). Find the number of possible arrangements if all the houses in styles \(A\) and \(D\) are in the first group and all the houses in styles \(B , C\) and \(E\) are in the second group.
3. Four of the 12 houses will be selected for a survey. Exactly one house must be in style \(B\) and exactly one house in style \(C\). Find the number of ways in which these four houses can be selected.

5

1. Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6
   1. if no digits are repeated,
   2. if repeated digits are allowed.
2. Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
   1. if there are more girls than boys in the team,
   2. if three of the boys are cousins and are either all in the team or all not in the team.

4

1. 1. Find how many different four-digit numbers can be made using only the digits 1, 3, 5 and 6 with no digit being repeated.
   2. Find how many different odd numbers greater than 500 can be made using some or all of the digits \(1,3,5\) and 6 with no digit being repeated.
2. Six cards numbered 1,2,3,4,5,6 are arranged randomly in a line. Find the probability that the cards numbered 4 and 5 are not next to each other.

6 \includegraphics[max width=\textwidth, alt={}, center]{fcf7b1c6-cc76-4c84-998c-9de6a7e9bb2d-3_163_618_260_765} Pegs are to be placed in the four holes shown, one in each hole. The pegs come in different colours and pegs of the same colour are identical. Calculate how many different arrangements of coloured pegs in the four holes can be made using

1. 6 pegs, all of different colours,
2. 4 pegs consisting of 2 blue pegs, 1 orange peg and 1 yellow peg. Beryl has 12 pegs consisting of 2 red, 2 blue, 2 green, 2 orange, 2 yellow and 2 black pegs. Calculate how many different arrangements of coloured pegs in the 4 holes Beryl can make using
3. 4 different colours,
4. 3 different colours,
5. any of her 12 pegs.

7 A committee of 6 people, which must contain at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women.

1. Find the number of possible committees that can be chosen.
2. Find the probability that one particular man, Albert, and one particular woman, Tracey, are both on the committee.
3. Find the number of possible committees that include either Albert or Tracey but not both.
4. The committee that is chosen consists of 4 men and 2 women. They queue up randomly in a line for refreshments. Find the probability that the women are not next to each other in the queue.

5

1. A team of 3 boys and 3 girls is to be chosen from a group of 12 boys and 9 girls to enter a competition. Tom and Henry are two of the boys in the group. Find the number of ways in which the team can be chosen if Tom and Henry are either both in the team or both not in the team.
2. The back row of a cinema has 12 seats, all of which are empty. A group of 8 people, including Mary and Frances, sit in this row. Find the number of different ways they can sit in these 12 seats if
   1. there are no restrictions,
   2. Mary and Frances do not sit in seats which are next to each other,
   3. all 8 people sit together with no empty seats between them.

6

1. A chess team of 2 girls and 2 boys is to be chosen from the 7 girls and 6 boys in the chess club. Find the number of ways this can be done if 2 of the girls are twins and are either both in the team or both not in the team.
2. 1. The digits of the number 1244687 can be rearranged to give many different 7-digit numbers. How many of these 7 -digit numbers are even?
   2. How many different numbers between 20000 and 30000 can be formed using 5 different digits from the digits \(1,2,4,6,7,8\) ?
3. Helen has some black tiles, some white tiles and some grey tiles. She places a single row of 8 tiles above her washbasin. Each tile she places is equally likely to be black, white or grey. Find the probability that there are no tiles of the same colour next to each other.

6 A shop has 7 different mountain bicycles, 5 different racing bicycles and 8 different ordinary bicycles on display. A cycling club selects 6 of these 20 bicycles to buy.

1. How many different selections can be made if there must be no more than 3 mountain bicycles and no more than 2 of each of the other types of bicycle? The cycling club buys 3 mountain bicycles, 1 racing bicycle and 2 ordinary bicycles and parks them in a cycle rack, which has a row of 10 empty spaces.
2. How many different arrangements are there in the cycle rack if the mountain bicycles are all together with no spaces between them, the ordinary bicycles are both together with no spaces between them and the spaces are all together?
3. How many different arrangements are there in the cycle rack if the ordinary bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?

6 The 11 letters of the word REMEMBRANCE are arranged in a line.

1. Find the number of different arrangements if there are no restrictions.
2. Find the number of different arrangements which start and finish with the letter M .
3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.