5.01a Permutations and combinations: evaluate probabilities

2 Fahad has 4 different coloured pairs of shoes (white, red, blue and black), 3 different coloured pairs of jeans (blue, black and brown) and 7 different coloured tee shirts (red, orange, yellow, blue, green, white and purple).

1. Fahad chooses an outfit consisting of one pair of shoes, one pair of jeans and one tee shirt. How many different outfits can he choose?
2. How many different ways can Fahad arrange his 3 jeans and 7 tee shirts in a row if the two blue items are not next to each other? Fahad also has 9 different books about sport. When he goes on holiday he chooses at least one of these books to take with him.
3. How many different selections are there if he can take any number of books ranging from just one of them to all of them?

7

1. Seven friends together with their respective partners all meet up for a meal. To commemorate the occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
   1. How many different arrangements are there if each friend is standing next to his or her partner?
   2. How many different arrangements are there if the 7 friends all stand together and the 7 partners all stand together?
2. A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be chosen if
   1. both boys are in the team,
   2. the adults are either all in the team or all not in the team,
   3. at least 2 girls are in the team?

3

1. In how many ways can all 9 letters of the word TELEPHONE be arranged in a line if the letters P and L must be at the ends? How many different selections of 4 letters can be made from the 9 letters of the word TELEPHONE if
2. there are no Es,
3. there is exactly 1 E ,
4. there are no restrictions?

6 Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr O'Connor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son. The 14 people all have to go through a turnstile one at a time to enter the theme park.

1. In how many different orders can the 14 people go through the turnstile if each family stays together?
2. In how many different orders can the 8 children and 6 adults go through the turnstile if no two adults go consecutively? Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
3. In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?

4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.

1. Find the probability that at least 2 of the 5 integers are less than or equal to 4 . Robert now generates \(n\) random integers between 1 and 9 inclusive. The random variable \(X\) is the number of these \(n\) integers which are less than or equal to a certain integer \(k\) between 1 and 9 inclusive. It is given that the mean of \(X\) is 96 and the variance of \(X\) is 32 .
2. Find the values of \(n\) and \(k\).

6 A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.

1. How many different selections of 12 trees can be made if there must be at least 2 of each type of tree? The council buys 4 hibiscus trees, 6 jacaranda trees and 2 oleander trees.
2. How many different arrangements of these 12 trees can be made if the hibiscus trees have to be next to each other, the jacaranda trees have to be next to each other and the oleander trees have to be next to each other?
3. How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another hibiscus tree?

2 The 12 houses on one side of a street are numbered with even numbers starting at 2 and going up to 24 . A free newspaper is delivered on Monday to 3 different houses chosen at random from these 12. Find the probability that at least 2 of these newspapers are delivered to houses with numbers greater than 14.

7 There are 10 spaniels, 14 retrievers and 6 poodles at a dog show. 7 dogs are selected to go through to the final.

1. How many selections of 7 different dogs can be made if there must be at least 1 spaniel, at least 2 retrievers and at least 3 poodles? 2 spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line.
2. How many different arrangements of these 7 dogs are there if the spaniels stand together and the retrievers stand together?
3. How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?

6 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that

1. all the letters A are together,
2. the first letter is a consonant ( \(\mathrm { T } , \mathrm { N } , \mathrm { Z }\) ), the second letter is a vowel ( \(\mathrm { A } , \mathrm { I }\) ), the third letter is a consonant, the fourth letter is a vowel, and so on alternately. 4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain
3. exactly 1 N and 1 A ,
4. exactly 1 N ?

2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.

5 Find how many different numbers can be made from some or all of the digits of the number 1345789 if

1. all seven digits are used, the odd digits are all together and no digits are repeated,
2. the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
3. the numbers made are multiples of 5 which are less than 1000 , and digits can be repeated.

3 A pet shop has 6 rabbits and 3 hamsters. 5 of these pets are chosen at random. The random variable \(X\) represents the number of hamsters chosen.

1. Show that the probability that exactly 2 hamsters are chosen is \(\frac { 10 } { 21 }\).
2. Draw up the probability distribution table for \(X\).

7 Nine cards are numbered \(1,2,2,3,3,4,6,6,6\).

1. All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way
   1. if the even digits are all together,
   2. if the first and last digits are both odd.
   3. Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way
      (a) if there are no repeated digits,
      (b) if the number is between 200 and 300 .

7

1. Find how many different numbers can be made by arranging all nine digits of the number 223677888 if
   1. there are no restrictions,
   2. the number made is an even number.
2. Sandra wishes to buy some applications (apps) for her smartphone but she only has enough money for 5 apps in total. There are 3 train apps, 6 social network apps and 14 games apps available. Sandra wants to have at least 1 of each type of app. Find the number of different possible selections of 5 apps that Sandra can choose.

4 A pet shop has 9 rabbits for sale, 6 of which are white. A random sample of two rabbits is chosen without replacement.

1. Show that the probability that exactly one of the two rabbits in the sample is white is \(\frac { 1 } { 2 }\).
2. Construct the probability distribution table for the number of white rabbits in the sample.
3. Find the expected value of the number of white rabbits in the sample.

7 Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.

1. She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make? Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if
2. she has a duck at each end of the row and no ducks anywhere else,
3. she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between.

6

1. 1. Find how many numbers there are between 100 and 999 in which all three digits are different.
   2. Find how many of the numbers in part (i) are odd numbers greater than 700 .
2. A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of 7 flowers from a shop to give to a friend. There must be at least 2 of each type of flower. The shop has 6 roses, 5 tulips and 4 daffodils, all different from each other. Find the number of different bunches of flowers that are possible.

7

1. Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if
   1. there are no restrictions,
   2. there are exactly six letters between the two Ws.
2. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.

6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if

1. there are no restrictions,
2. the first letter is R and the last letter is G ,
3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
4. Find the number of selections which contain no Es and exactly 1 R .
5. Find the number of selections which contain no Es.

7

1. Eight children of different ages stand in a random order in a line. Find the number of different ways this can be done if none of the three youngest children stand next to each other.
2. David chooses 5 chocolates from 6 different dark chocolates, 4 different white chocolates and 1 milk chocolate. He must choose at least one of each type. Find the number of different selections he can make.
3. A password for Chelsea's computer consists of 4 characters in a particular order. The characters are chosen from the following.
   The password must include at least one capital letter, at least one digit and at least one symbol. No character can be repeated. Find the number of different passwords that Chelsea can make.

6 A library contains 4 identical copies of book \(A , 2\) identical copies of book \(B\) and 5 identical copies of book \(C\). These 11 books are arranged on a shelf in the library.

1. Calculate the number of different arrangements if the end books are either both book \(A\) or both book \(B\).
2. Calculate the number of different arrangements if all the books \(A\) are next to each other and none of the books \(B\) are next to each other.

6

1. Find how many numbers between 3000 and 5000 can be formed from the digits \(1,2,3,4\) and 5,
   1. if digits are not repeated,
   2. if digits can be repeated and the number formed is odd.
2. A box of 20 biscuits contains 4 different chocolate biscuits, 2 different oatmeal biscuits and 14 different ginger biscuits. 6 biscuits are selected from the box at random.
   1. Find the number of different selections that include the 2 oatmeal biscuits.
   2. Find the probability that fewer than 3 chocolate biscuits are selected.

7 Find the number of different ways in which all 9 letters of the word MINCEMEAT can be arranged in each of the following cases.

1. There are no restrictions.
2. No vowel (A, E, I are vowels) is next to another vowel.
   5 of the 9 letters of the word MINCEMEAT are selected.
3. Find the number of possible selections which contain exactly 1 M and exactly 1 E .
4. Find the number of possible selections which contain at least 1 M and at least 1 E .
   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 Mrs Rupal chooses 3 animals at random from 5 dogs and 2 cats. The random variable \(X\) is the number of cats chosen.

1. Draw up the probability distribution table for \(X\).
2. You are given that \(\mathrm { E } ( X ) = \frac { 6 } { 7 }\). Find the value of \(\operatorname { Var } ( X )\).

6

1. Find the number of ways in which all 9 letters of the word AUSTRALIA can be arranged in each of the following cases.
   1. All the vowels (A, I, U are vowels) are together.
   2. The letter T is in the central position and each end position is occupied by one of the other consonants (R, S, L).
2. Donna has 2 necklaces, 8 rings and 4 bracelets, all different. She chooses 4 pieces of jewellery. How many possible selections can she make if she chooses at least 1 necklace and at least 1 bracelet?