5.01b Selection/arrangement: probability problems

5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.

1. In how many ways can the group of 7 be chosen?
   On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
2. In how many ways can the group of 5 be chosen?

6

1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-10_2715_35_110_2012}
3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
   The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
4. Find the probability that both Ps are in one group and both Ss are in the other group.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 A committee of 5 people is to be chosen from 6 men and 4 women. In how many ways can this be done

1. if there must be 3 men and 2 women on the committee,
2. if there must be more men than women on the committee,
3. if there must be 3 men and 2 women, and one particular woman refuses to be on the committee with one particular man?

7

1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
   1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
   2. if exactly one forward player must be included, together with any two others.
2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
   1. if there are no restrictions on the order of the letters,
   2. if the two letters D come first and the two letters L come last.

4 \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733} The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.

1. How many possible seating arrangements are there for the 11 passengers?
2. How many possible seating arrangements are there if 5 particular people sit in the back row? Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.
3. In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?

5

1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
   1. if there are no restrictions,
   2. if the Rs must all be together.
   3. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?

3 Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.

1. How many different arrangements of all 11 CDs on a shelf are there if the jazz CDs are all next to each other?
2. Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different possible selections can be made?

4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.

1. In how many different ways can the group be chosen?
2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?

6

1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.

4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

1. the first and last cans in the row are the same type of drink,
2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.

4

1. Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in a line.
2. The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability that the 3 vowels (A, E, I) are not all next to each other.
3. Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss.

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?

5 An English examination consists of 8 questions in Part \(A\) and 3 questions in Part \(B\). Candidates must choose 6 questions. The order in which questions are chosen does not matter. Find the number of ways in which the 6 questions can be chosen in each of the following cases.

1. There are no restrictions on which questions can be chosen.
2. Candidates must choose at least 4 questions from Part \(A\).
3. Candidates must either choose both question 1 and question 2 in Part \(A\), or choose neither of these questions.

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?

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?

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.

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.