5.01b Selection/arrangement: probability problems

2 A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.

4 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.

1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_284_467_1491_495} \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_286_471_1489_1183} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
2. Find the number of different seating arrangements that are now possible for the 8 friends.

6

1. A village hall has seats for 40 people, consisting of 8 rows with 5 seats in each row. Mary, Ahmad, Wayne, Elsie and John are the first to arrive in the village hall and no seats are taken before they arrive.
   1. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John assuming there are no restrictions?
   2. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John if Mary and Ahmad sit together in the front row and the other three sit together in one of the other rows?
2. In how many ways can a team of 4 people be chosen from 10 people if 2 of the people, Ross and Lionel, refuse to be in the team together?

6

1. Find the number of different 3-digit numbers greater than 300 that can be made from the digits \(1,2,3,4,6,8\) if
   1. no digit can be repeated,
   2. a digit can be repeated and the number made is even.
2. A team of 5 is chosen from 6 boys and 4 girls. Find the number of ways the team can be chosen if
   1. there are no restrictions,
   2. the team contains more boys than girls.

6 A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.

1. Find the number of possible arrangements of the 5 cars in the car park.
2. Find the probability that the 5 cars are not all next to each other.
   On another day, 12 cars of different makes are parked in the car park. 5 of these cars are red, 4 are white and 3 are black. Elizabeth selects 3 of these cars.
   [0pt]
3. Find the number of selections Elizabeth can make that include cars of at least 2 different colours. [5]

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]

4 Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.

1. Find the probability that the group of 7 includes one particular boy.
2. Find the probability that the group of 7 includes at least 2 girls.

7

1. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.
2. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.
3. Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.
4. Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T .
   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 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 ?

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.

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.

4 Mary saves her digital images on her computer in three separate folders named 'Family', 'Holiday' and 'Friends'. Her family folder contains 3 images, her holiday folder contains 4 images and her friends folder contains 8 images. All the images are different.

1. Find in how many ways she can arrange these 15 images in a row across her computer screen if she keeps the images from each folder together.
2. Find the number of different ways in which Mary can choose 6 of these images if there are 2 from each folder.
3. Find the number of different ways in which Mary can choose 6 of these images if there are at least 3 images from the friends folder and at least 1 image from each of the other two folders.

7

1. In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together.
2. Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.
   1. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account? The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.
   2. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen?
3. 9 different fruit pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done.

3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.

1. How many different arrangements of the letters are possible?
2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
3. Find the probability that at least one of these 2 cards has D printed on it.

3

1. How many different teams of 7 people can be chosen, without regard to order, from a squad of 15 ?
2. The squad consists of 6 forwards and 9 defenders. How many different teams containing 3 forwards and 4 defenders can be chosen?

3 Five friends, Ali, Bev, Carla, Don and Ed, stand in a line for a photograph.

1. How many different possible arrangements are there if Ali, Bev and Carla stand next to each other?
2. How many different possible arrangements are there if none of Ali, Bev and Carla stand next to each other?
3. If all possible arrangements are equally likely, find the probability that two of Ali, Bev and Carla are next to each other, but the third is not next to either of the other two.

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

1. In how many ways can the manager select the forwards?
2. In how many ways can the manager select the team?

6 A band has a repertoire of 12 songs suitable for a live performance. From these songs, a selection of 7 has to be made.

1. Calculate the number of different selections that can be made.
2. Once the 7 songs have been selected, they have to be arranged in playing order. In how many ways can this be done?

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.

4 An examination paper consists of three sections.

• Section A contains 6 questions of which the candidate must answer 3
• Section B contains 7 questions of which the candidate must answer 4
• Section C contains 8 questions of which the candidate must answer 5
  1. In how many ways can a candidate choose 3 questions from Section A?
  2. In how many ways can a candidate choose 3 questions from Section A, 4 from Section B and 5 from Section C?

A candidate does not read the instructions and selects 12 questions at random.

Find the probability that they happen to be 3 from Section A, 4 from Section B and 5 from Section C.

4 Peter and Esther visit a restaurant for a three-course meal. On the menu there are 4 starters, 5 main courses and 3 sweets. Peter and Esther each order a starter, a main course and a sweet.

1. Calculate the number of ways in which Peter may choose his three-course meal.
2. Suppose that Peter and Esther choose different dishes from each other.
   (A) Show that the number of possible combinations of starters is 6 .
   (B) Calculate the number of possible combinations of 6 dishes for both meals.
3. Suppose instead that Peter and Esther choose their dishes independently.
   (A) Write down the probability that they choose the same main course.
   (B) Find the probability that they choose different dishes from each other for every course.

1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.

1. In how many ways can she select the 4 tracks?
2. In how many different orders can she arrange the 4 tracks once she has chosen them?

3 At a dog show, three out of eleven dogs are to be selected for a national competition.

1. Find the number of possible selections.
2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.

6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.

1. (a) How many different arrangements are possible?
   (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
2. (a) How many different arrangements are possible?
   (b) Find the probability that questions A and H are next to each other.
   (c) Find the probability that questions B and J are separated by more than four other questions.

9 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 .

1. Andrea chooses 4 discs at random, without replacement, and places them in a row.
   1. How many different 4 -digit numbers can be made?
   2. How many different odd 4-digit numbers can be made?
   3. Andrea's 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that
      (a) the 4 digits include at least 3 odd digits,
      (b) the 4 digits add up to 28 .