5.01b Selection/arrangement: probability problems

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.

1 A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue. How many different selections of 5 marbles contain at least 4 marbles of the same colour?

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?

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}

1 The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car. In how many ways can the members who will travel in the coach be chosen?

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.

5 A group of 12 people consists of 3 boys, 4 girls and 5 adults.

1. In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
2. In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?
   The same group of 12 people stand in a line.
3. How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?

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.

6 A Social Club has 15 members, of whom 8 are men and 7 are women. The committee of the club consists of 5 of its members.

1. Find the number of different ways in which the committee can be formed from the 15 members if it must include more men than women.
   The 15 members are having their photograph taken. They stand in three rows, with 3 people in the front row, 5 people in the middle row and 7 people in the back row.
2. In how many different ways can the 15 members of the club be divided into a group of 3, a group of 5 and a group of 7 ?
   In one photograph Abel, Betty, Cally, Doug, Eve, Freya and Gino are the 7 members in the back row.
3. In how many different ways can these 7 members be arranged so that Abel and Betty are next to each other and Freya and Gino are not 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 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.
2. The 9 letters in the word ALLIGATOR are arranged in a random order. Find the probability that the two Ls are together and there are exactly 6 letters between the two As.
3. Find the number of different selections of 5 letters from the 9 letters in the word ALLIGATOR which contain at least one A 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.

6

1. Find the number of different arrangements of the 9 letters in the word ACTIVATED.
2. Find the number of different arrangements of the 9 letters in the word ACTIVATED in which there are at least 5 letters between the two As.
   Five letters are selected at random from the 9 letters in the word ACTIVATED.
3. Find the probability that the selection does not contain more Ts than As.

6 \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-12_291_809_255_667} In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, \(X\) and \(Y\). Rajid, Sue and Tan are three of these friends.

1. The eight friends will be divided into two groups of 4, one group for table \(X\) and one group for table \(Y\). Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.
   When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table \(X\) on the same side as each other. Tan decides that he does not mind at which table he sits.
2. Find the number of different seating arrangements for the 8 friends.
   As they leave the restaurant, the 8 friends stand in a line for a photograph.
3. Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.
   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 9 letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)
2. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together.
   Four letters are selected at random from the 9 letters in the word ANDROMEDA.
3. Find the probability that this selection contains at least one D and exactly one A .
   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 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.

1. At the beginning of the party, the 12 people will stand in a line for a photograph.
   1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
   2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as 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. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
   Four letters are selected at random from the 9 letters in the word INTELLECT.
   [0pt]
3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}

2

1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}