5.01a Permutations and combinations: evaluate probabilities

6

1. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.
2. Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.
3. Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E .

7

1. Find the number of different possible arrangements of the 9 letters in the word CELESTIAL.
2. Find the number of different arrangements of the 9 letters in the word CELESTIAL in which the first letter is C, the fifth letter is T and the last letter is E.
3. Find the probability that a randomly chosen arrangement of the 9 letters in the word CELESTIAL does not have the two Es together.
   5 letters are selected at random from the 9 letters in the word CELESTIAL.
4. Find the number of different selections if the 5 letters include at least one E and at most one L .
   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

1. How many different arrangements are there of the 8 letters in the word RELEASED?
2. How many different arrangements are there of the 8 letters in the word RELEASED in which the letters LED appear together in that order?
3. An arrangement of the 8 letters in the word RELEASED is chosen at random. Find the probability that the letters A and D are not together.

6

1. Find the total number of different arrangements of the 8 letters in the word TOMORROW.
2. Find the total number of different arrangements of the 8 letters in the word TOMORROW that have an R at the beginning and an R at the end, and in which the three Os are not all together.
   Four letters are selected at random from the 8 letters of the word TOMORROW.
3. Find the probability that the selection contains at least one O and at least one R .

6

1. How many different arrangements are there of the 11 letters in the word REQUIREMENT? [2]
2. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which the two Rs are together and the three Es are together?
3. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which there are exactly three letters between the two Rs?
   Five of the 11 letters in the word REQUIREMENT are selected.
4. How many possible selections contain at least two Es and at least one R?

1

1. Find the number of different arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.
2. Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.

2 There are 6 men and 8 women in a Book Club. The committee of the club consists of five of its members. Mr Lan and Mrs Lan are members of the club.

1. In how many different ways can the committee be selected if exactly one of Mr Lan and Mrs Lan must be on the committee?
2. In how many different ways can the committee be selected if Mrs Lan must be on the committee and there must be more women than men on the committee?

6 Sajid is practising for a long jump competition. He counts any jump that is longer than 6 m as a success. On any day, the probability that he has a success with his first jump is 0.2 . For any subsequent jump, the probability of a success is 0.3 if the previous jump was a success and 0.1 otherwise. Sajid makes three jumps.

1. Draw a tree diagram to illustrate this information, showing all the probabilities.
2. Find the probability that Sajid has exactly one success given that he has at least one success.
   On another day, Sajid makes six jumps.
3. Find the probability that only his first three jumps are successes or only his last three jumps are successes.

7 A group of 15 friends visit an adventure park. The group consists of four families.

• Mr and Mrs Kenny and their four children
• Mr and Mrs Lizo and their three children
• Mrs Martin and her child
• Mr and Mrs Nantes

The group travel to the park in three cars, one containing 6 people, one containing 5 people and one containing 4 people. The cars are driven by Mr Lizo, Mrs Martin and Mr Nantes respectively.

1. In how many different ways can the remaining 12 members of the group be divided between the three cars?
   The group enter the park by walking through a gate one at a time.
2. In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?
   In the park, the group enter a competition which requires a team of 4 adults and 3 children.
3. In how many ways can the team be chosen from the group of 15 so that the 3 children are all from different families?
4. In how many ways can the team be chosen so that at least one of Mr Kenny or Mr Lizo is included?
   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. Find the number of ways in which a committee of 6 people can be chosen from 6 men and 8 women if it must include 3 men and 3 women.
   A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
2. Find the number of ways in which this committee can be chosen if there are no restrictions on the numbers of men and women, but it must include no more than two of the brothers.

3

1. Find the number of different arrangements of the 8 letters in the word COCOONED.
2. Find the number of different arrangements of the 8 letters in the word COCOONED in which the first letter is O and the last letter is N .
3. Find the probability that a randomly chosen arrangement of the 8 letters in the word COCOONED has all three Os together given that the two Cs are next to each other.

7 A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.

1. Find the probability that the first time that the free gift is an elephant is before the 6th Monday.
2. Find the probability that Sahim will get more than two leopards in the 12 magazines.
3. Find the probability that after 5 weeks Sahim has exactly one of each animal.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition.

1. Find the number of different ways in which the team of 7 can be selected if it consists of exactly 1 swimmer, at least 4 cyclists and at most 2 runners.
   For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph.
2. How many different arrangements are there of the 9 people if the swimmers stand together, the cyclists stand together and the runners stand together?
3. How many different arrangements are there of the 9 people if none of the cyclists stand next to each other?
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Find the number of different arrangements of the 10 letters in the word CASABLANCA in which the two Cs are not together.
2. Find the number of different arrangements of the 10 letters in the word CASABLANCA which have an A at the beginning, an A at the end and exactly 3 letters between the 2 Cs .
   Five letters are selected from the 10 letters in the word CASABLANCA.
3. Find the number of different selections in which the five letters include at least two As and at most one C.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The eight digits \(1,2,2,3,4,4,4,5\) are arranged in a line.

1. How many different arrangements are there of these 8 digits?
2. Find the number of different arrangements of the 8 digits in which there is a 2 at the beginning, a 2 at the end and the three 4 s are not all together.
   Three digits are selected at random from the eight digits \(1,2,2,3,4,4,4,5\).
3. Find the probability that the three digits are all different.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. How many different arrangements are there of the 10 letters in the word REGENERATE?
2. How many different arrangements are there of the 10 letters in the word REGENERATE in which the 4 Es are together and the 2 Rs have exactly 3 letters in between them?
3. Find the probability that a randomly chosen arrangement of the 10 letters in the word REGENERATE is one in which the consonants ( \(\mathrm { G } , \mathrm { N } , \mathrm { R } , \mathrm { R } , \mathrm { T }\) ) and vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { E } , \mathrm { E }\) ) alternate, so that no two consonants are next to each other and no two vowels are next to each other. [5]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6

1. How many different arrangements are there of the 9 letters in the word RECORDERS?
2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together? \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
   The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
3. Find the probability that the three Rs are in the same group.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}

4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.

1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.

6

1. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
2. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
3. Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.

6 A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.

1. In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?
   The 10 members of the club stand in a line for a photograph.
2. How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?
   For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club.
3. How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?
   If you use the following page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that all 3 Es are together.
2. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that the Ps are not next to each other.
3. Find the probability that a randomly chosen arrangement of the 10 letters of the word SHOPKEEPER has an E at the beginning and an E at the end.
   Four letters are selected from the 10 letters of the word SHOPKEEPER.
4. Find the number of different selections if the four letters include exactly one P .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Mr and Mrs Ahmed with their two children, and Mr and Mrs Baker with their three children, are visiting an activity centre together. They will divide into groups for some of the activities.

1. In how many ways can the 9 people be divided into a group of 6 and a group of 3?
   5 of the 9 people are selected at random for a particular activity.
2. Find the probability that this group of 5 people contains all 3 of the Baker children.
   All 9 people stand in a line.
3. Find the number of different arrangements in which Mr Ahmed is not standing next to Mr Baker.
4. Find the number of different arrangements in which there is exactly one person between Mr Ahmed and Mr Baker.
   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 committee of 6 people is to be chosen from 9 women and 5 men.

1. Find the number of ways in which the 6 people can be chosen if there must be more women than men on the committee.
   The 9 women and 5 men include a sister and brother.
2. Find the number of ways in which the committee can be chosen if the sister and brother cannot both be on the committee.

5 The 8 letters in the word RESERVED are arranged in a random order.

1. Find the probability that the arrangement has V as the first letter and E as the last letter.
2. Find the probability that the arrangement has both Rs together given that all three Es are together.

5 Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken. For the first photograph, the 9 members will stand in a line.

1. How many different arrangements of the 9 members are possible in which Raman will be at the centre of the line?
2. How many different arrangements of the 9 members are possible in which Raman and Sanjay are not next to each other?
   For the second photograph, the members will stand in two rows, with 5 in the back row and 4 in the front row.
3. In how many different ways can the 9 members be divided into a group of 5 and a group of 4?
4. For a random division into a group of 5 and a group of 4, find the probability that Raman and Sanjay are in the same group as each other.