Multi-stage selection problems

7 A group of 15 friends visit an adventure park. The group consists of four families.

• Mr and Mrs Kenny and their four children
• Mr and Mrs Lizo and their three children
• Mrs Martin and her child
• Mr and Mrs Nantes

The group travel to the park in three cars, one containing 6 people, one containing 5 people and one containing 4 people. The cars are driven by Mr Lizo, Mrs Martin and Mr Nantes respectively.

1. In how many different ways can the remaining 12 members of the group be divided between the three cars?
   The group enter the park by walking through a gate one at a time.
2. In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?
   In the park, the group enter a competition which requires a team of 4 adults and 3 children.
3. In how many ways can the team be chosen from the group of 15 so that the 3 children are all from different families?
4. In how many ways can the team be chosen so that at least one of Mr Kenny or Mr Lizo is included?
   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 In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition.

1. Find the number of different ways in which the team of 7 can be selected if it consists of exactly 1 swimmer, at least 4 cyclists and at most 2 runners.
   For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph.
2. How many different arrangements are there of the 9 people if the swimmers stand together, the cyclists stand together and the runners stand together?
3. How many different arrangements are there of the 9 people if none of the cyclists stand 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.

5 A group of 12 people consists of 3 boys, 4 girls and 5 adults.

1. In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
2. In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?
   The same group of 12 people stand in a line.
3. How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?

6 A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.

1. In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?
   The 10 members of the club stand in a line for a photograph.
2. How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?
   For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club.
3. How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?
   If you use the following page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Mr and Mrs Ahmed with their two children, and Mr and Mrs Baker with their three children, are visiting an activity centre together. They will divide into groups for some of the activities.

1. In how many ways can the 9 people be divided into a group of 6 and a group of 3?
   5 of the 9 people are selected at random for a particular activity.
2. Find the probability that this group of 5 people contains all 3 of the Baker children.
   All 9 people stand in a line.
3. Find the number of different arrangements in which Mr Ahmed is not standing next to Mr Baker.
4. Find the number of different arrangements in which there is exactly one person between Mr Ahmed and Mr Baker.
   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 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.

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.

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.

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.

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?

3 In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.

1. How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist and no more than 1 double bass player?
   The small group that is selected contains 4 violinists, 1 cellist and 1 double bass player. They sit in a line to perform a concert.
   [0pt]
2. How many different arrangements are there of these 6 musicians if the violinists must sit together? [3]

3 A sports team of 7 people is to be chosen from 6 attackers, 5 defenders and 4 midfielders. The team must include at least 3 attackers, at least 2 defenders and at least 1 midfielder.

1. In how many different ways can the team of 7 people be chosen?
   The team of 7 that is chosen travels to a match in two cars. A group of 4 travel in one car and a group of 3 travel in the other car.
2. In how many different ways can the team of 7 be divided into a group of 4 and a group of 3 ?

3

1. Geoff wishes to plant 25 flowers in a flower-bed. He can choose from 15 different geraniums, 10 different roses and 8 different lilies. He wants to have at least 11 geraniums and also to have the same number of roses and lilies. Find the number of different selections of flowers he can make.
2. Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged if exactly two of the Gs are next to each other.

8 A group of 8 people, including Kathy, David and Harpreet, are planning a theatre trip.

1. Four of the group are chosen at random, without regard to order, to carry the refreshments. Find the probability that these 4 people include Kathy and David but not Harpreet.
2. The 8 people sit in a row. Kathy and David sit next to each other and Harpreet sits at the left-hand end of the row. How many different arrangements of the 8 people are possible?
3. The 8 people stand in a line to queue for the exit. Kathy and David stand next to each other and Harpreet stands next to them. How many different arrangements of the 8 people are possible?

6 A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher chooses 5 of the books at random.

1. Find the probability that 3 of the books are on Algebra. The teacher now arranges all 10 books in random order on a shelf.
2. Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.}
3. Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.

3 A para relay team of 4 swimmers needs to be chosen from a group of 7 swimmers.

1. How many ways are there to choose 4 swimmers from 7? There are no restrictions on how many men and how many women are in the team. The group of 7 swimmers consists of 5 men and 2 women.
2. How many ways are there to choose a team with more men than women? The physical impairment of each swimmer is given a score.
   The scores for the swimmers are \(\begin{array} { l l l l l l l } 3 & 4 & 4 & 6 & 7 & 8 & 9 \end{array}\) The total score for the team must be 20 or less.
3. How many different valid teams are possible? The order of the swimmers in the team is now taken into consideration.
4. In total, how many different arrangements are there of valid teams?
5. In how many of these valid teams are the scores of the swimmers in increasing order? For example, 3, 4, 4, 8 but not 4, 3, 4, 8 .

4 The first 20 consecutive positive integers include the 8 prime numbers \(2,3,5,7,11,13,17\) and 19. Emma randomly chooses 5 distinct numbers from the first 20 consecutive positive integers. The order in which Emma chooses the numbers does not matter.

1. Calculate the number of possibilities in which Emma's 5 numbers include exactly 2 prime numbers and 3 non-prime numbers.
2. Calculate the number of possibilities in which Emma's 5 numbers include at least 2 prime numbers. The pairs \(\{ 3,13 \}\) and \(\{ 7,17 \}\) each consist of numbers with a difference of exactly 10 .
3. Calculate the number of possibilities in which Emma's 5 numbers include at least one pair of prime numbers in which the difference between them is exactly 10 . A new set of 20 consecutive positive integers, each with at least two digits, is chosen. This set of 20 numbers contains 5 prime numbers.
4. Use the pigeonhole principle to show that there is at least one pair of these prime numbers for which the difference between them is exactly 10 .

2 A group of 6 people is to be chosen from 4 men and 11 women.

1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
   Two of the 11 women are sisters Jane and Kate.
2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?

Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.

1. How many different ways are there to pair the letters with the envelopes? [1]
2. Find the number of arrangements with exactly three letters in the correct envelopes. [2]
3. 1. Show that there are two derangements of the three symbols A, B and C. [1]
   2. Hence find the number of arrangements with exactly two letters in the correct envelopes. [1]

Let \(D_n\) represent the number of derangements of \(n\) symbols.

1. Explain why \(D_n = (n-1) \times (D_{n-1} + D_{n-2})\). [2]
2. Find the number of ways in which all five letters are in the wrong envelopes. [2]