5.01b Selection/arrangement: probability problems

5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits \(1,1,2,3,4,5,6\) on them.

1. How many different 7-digit numbers can be made by arranging Eleanor's cards?
2. Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?

6 A volleyball squad has 11 players. A volleyball team consists of 6 players.

1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.

1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]

Five letters are selected from the 9 letters in the word DELIVERED.

1. Find the number of different selections if the 5 letters include at least one D and at least one E. [3]

The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included? [3]

Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order.

1. Find the number of possible selections of 3 letters. [2]
2. Find the probability that the letter P is included in the selection. [3]

Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line.

1. Find the probability that the 3 letters form the word TOP. [3]

Three skaters, \(A\), \(B\) and \(C\), are placed in rank order by four judges. Judge \(P\) ranks skater \(A\) in 1st place, skater \(B\) in 2nd place and skater \(C\) in 3rd place.

1. Without carrying out any calculation, state the value of Spearman's rank correlation coefficient for the following ranks. Give a reason for your answer. [1]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(Q\)	3	2	1

2. Calculate the value of Spearman's rank correlation coefficient for the following ranks. [3]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(R\)	3	1	2

3. Judge \(S\) ranks the skaters at random. Find the probability that the value of Spearman's rank correlation coefficient between the ranks of judge \(P\) and judge \(S\) is 1. [3]

The menu below shows all the dishes available at a certain restaurant. A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,

1. find the number of possible combinations of dishes that they can choose to share, [3]
2. assuming that all choices are equally likely, find the probability that they choose boiled rice. [2]

The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.

1. Find the number of possible combinations of dishes that they can now choose. [3]

The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other. [3]
2. Find the probability that no two men are next to one another. [4]

Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than 1%. Determine the smallest possible value of \(n\). [7]

The letters of the word CHAFFINCH are written on cards.

1. In how many ways can the letters be rearranged with no restrictions. [1]
2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]

An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.

1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. [2]

Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark questions are the first four questions on the paper, but are arranged in random order. The 9-mark questions are the last four questions, but are arranged in random order. Find the probability that

1. the questions on geometric distributions and on binomial distributions are next to one another, [2]
2. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions. [3]

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
2. All 26 cards are arranged in a line, in random order.
   1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
   2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]

There are 15 students enrolled in a Maths club.

1. In how many ways is it possible to choose 4 of the students to take part in a competition? [2]

There are 4 different medals to be allocated, at random, to the students in the Maths club.

1. If there are no restrictions about how many medals a student may receive, in how many ways can the medals be allocated? [2]
2. Find the probability that no student receives more than one medal. [3]