5.01a Permutations and combinations: evaluate probabilities

6

1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.

7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done

1. if there are more women than men on the committee,
2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
3. In how many different ways can the committee members be arranged in a line if the man is not at either end?

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.

5

1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
   1. if all the letters E are together,
   2. if the T is at one end and there is an S at the other end.
2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .

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.

3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that

1. he gets a green robot on opening his first packet,
2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
3. Find the probability that the first four packets Amos opens all contain different coloured robots.

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.

5

1. Find the number of different ways that the 13 letters of the word ACCOMMODATION can be arranged in a line if all the vowels (A, I, O) are next to each other.
2. There are 7 Chinese, 6 European and 4 American students at an international conference. Four of the students are to be chosen to take part in a television broadcast. Find the number of different ways the students can be chosen if at least one Chinese and at least one European student are included.

5

1. Find the number of different ways of arranging all nine letters of the word PINEAPPLE if no vowel (A, E, I) is next to another vowel.
2. A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 allrounders and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicket-keeper.
   1. Find the number of different teams the manager can choose.
   2. Find the number of different teams the manager can choose if one particular batsman refuses to be in the team when one particular bowler is in the team.

6 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that

1. the vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { O }\) ) are together and the consonants ( \(\mathrm { C } , \mathrm { G } , \mathrm { H } , \mathrm { N } , \mathrm { P }\) ) are together, [3]
2. the Es are not next to each other. Four letters are selected from the 10 letters of the word COPENHAGEN.
3. Find the number of different selections if the four letters must contain the same number of Es and Ns with at least one of each.

1 A committee of 5 people is to be chosen from 4 men and 6 women. William is one of the 4 men and Mary is one of the 6 women. Find the number of different committees that can be chosen if William and Mary refuse to be on the committee together.

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.

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]

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.

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.

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.

6

1. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.
2. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.
   Four letters are selected from the 12 letters of the word STEEPLECHASE.
3. Find the number of different selections if the four letters include exactly one \(S\).

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.

2

1. How many different arrangements are there of the 9 letters in the word CORRIDORS?
2. How many different arrangements are there of the 9 letters in the word CORRIDORS in which the first letter is D and the last letter is R or O ?

6 \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-4_387_899_255_623} A small aeroplane has 14 seats for passengers. The seats are arranged in 4 rows of 3 seats and a back row of 2 seats (see diagram). 12 passengers board the aeroplane.

1. How many possible seating arrangements are there for the 12 passengers? Give your answer correct to 3 significant figures. These 12 passengers consist of 2 married couples (Mr and Mrs Lin and Mr and Mrs Brown), 5 students and 3 business people.
2. The 3 business people sit in the front row. The 5 students each sit at a window seat. Mr and Mrs Lin sit in the same row on the same side of the aisle. Mr and Mrs Brown sit in another row on the same side of the aisle. How many possible seating arrangements are there?
3. If, instead, the 12 passengers are seated randomly, find the probability that Mrs Lin sits directly behind a student and Mrs Brown sits in the front row.

6

1. Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be arranged
   1. if there are no restrictions,
   2. if the 4 vowels \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { I }\) must all be together.
2. 1. 4 astronauts are chosen from a certain number of candidates. If order of choosing is not taken into account, the number of ways the astronauts can be chosen is 3876 . How many ways are there if order of choosing is taken into account?
   2. 4 astronauts are chosen to go on a mission. Each of these astronauts can take 3 personal possessions with him. How many different ways can these 12 possessions be arranged in a row if each astronaut's possessions are kept together?

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.