Permutations and combinations

Ranuf and Saed are two of the 40 club members. Every one of the 40 members will go to a concert. 35 members will travel by coach and the other 5 will travel by car. Ranuf will be on the coach and Saed will be in the car.

Feb/March 2020

Richard owns $3$ blue candles, $2$ red candles and $6$ green candles. Except for colour, the candles are identical. He places the $11$ candles in a straight line.

Feb/March 2020

Find the total number of distinct arrangements of the $11$ letters in the word CATERPILLAR.

Feb/March 2021

The group of $12$ people is made up of $3$ boys, $4$ girls and $5$ adults.

Feb/March 2022

Find how many different arrangements of the $9$ letters in DELIVERED have the three Es together and the two Ds not adjacent.

Feb/March 2023

A village social club has just been formed and includes 10 members, comprising 6 men and 4 women. The committee of the club is to be made up of 5 members.

Feb/March 2024

Alissa owns $10$ books from the Squares and Circles series. Apart from their colours, the books all appear alike. The set contains $3$ blue books, $2$ red books, $2$ yellow books, $1$ orange book, $1$ purple book and $1$ green book. She lines the books up on a shelf, and she cares only about the colour sequence.

Feb/March 2025

Find the number of different arrangements that can be made from the 9 letters in the word JEWELLERY when the three Es are together and the two Ls are together.

May/June 2020

A music competition features $8$ pianists, $4$ guitarists and $6$ violinists. $7$ of these musicians are to be chosen to progress to the final.

May/June 2020

Determine the number of distinct arrangements of the 10 letters in SUMMERTIME that have an E at the beginning and an E at the end.

May/June 2020

Find how many distinct arrangements are possible using the $9$ letters in the word CELESTIAL.

May/June 2020

A bag contains 12 marbles, and no two are the same size. 8 of the marbles are red and 4 of the marbles are blue.

May/June 2021

How many distinct arrangements can be made from the $8$ letters in the word RELEASED?

May/June 2021

Determine how many different arrangements can be made from the 8 letters in TOMORROW.

May/June 2021

In how many distinct ways can the $11$ letters of REQUIREMENT be arranged?

May/June 2021

Determine the number of distinct arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.

May/June 2022

A Book Club contains 6 men and 8 women. The club’s committee is formed from five members. Mr Lan and Mrs Lan are both members of the club.

May/June 2022

Find the number of distinct arrangements of the $9$ letters in CROCODILE.

May/June 2022

A party of $15$ friends goes to an adventure park. There are four families in the party: - 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 They reach the park in three cars, with $6$ people in one car, $5$ people in another car and $4$ people in the third car. The cars are driven by Mr Lizo, Mrs Martin and Mr Nantes respectively.

May/June 2022

Determine how many different committees of 6 people can be formed from 6 men and 8 women when the committee has to contain 3 men and 3 women.

May/June 2023

Determine how many distinct arrangements can be made from the 8 letters in the word COCOONED.

May/June 2023

Among a set of $25$ people, $6$ are swimmers, $8$ are cyclists and $11$ are runners. Every person takes part in just one of these sports. A team of $7$ people is chosen from these $25$ people to enter a competition.

May/June 2023

Find how many different arrangements of the $10$ letters in CASABLANCA have the two Cs not adjacent.

May/June 2023

The eight digits $1, 2, 2, 3, 4, 4, 4, 5$ are placed in one row.

May/June 2024

What is the number of distinct arrangements of the $10$ letters in REGENERATE?

May/June 2024

In how many distinct ways can the 9 letters of RECORDERS be arranged?

May/June 2024

Among 20 musicians, 9 are guitarists, 6 are pianists and 5 are drummers. From these 20, 6 musicians are chosen to perform at a concert.

May/June 2025

Determine how many distinct arrangements of the 10 letters in AMALGAMATE there are when an M is at the start, another M is at the end, and no As are adjacent.

May/June 2025

A group of friends contains $7$ men and $4$ women. Among the men, three are brothers: Ali, Ben and Charlie.

May/June 2025

A darts club consists of 12 members: 7 men and 5 women. Each Monday, 4 people are selected at random to represent the club in a competition.

May/June 2025

Determine how many different arrangements can be made from the $10$ letters in SHOPKEEPER if all $3$ Es are kept together.

Oct/Nov 2020

Mr and Mrs Ahmed, together with their two children, and Mr and Mrs Baker, together with their three children, are going to an activity centre as a group. For some activities, they will be split into separate groups.

Oct/Nov 2020

A committee made up of 6 people is to be selected from 9 women and 5 men.

Oct/Nov 2020

The 8 letters in RESERVED are put into a random order.

Oct/Nov 2020

Raman and Sanjay are two of the $9$ people in a quiz team altogether. Two photographs of the quiz team are to be taken. In the first photograph, the $9$ members are arranged in a straight line.

Oct/Nov 2021

A set of $6$ people is to be selected from $4$ men and $11$ women.

Oct/Nov 2021

How many distinct arrangements are possible for the 9 letters in TELESCOPE?

Oct/Nov 2021

The local sports club has 26 members, including Mr and Mrs Khan and their son Abad. A party is being organised for Abad’s birthday, but the venue can accommodate only 20 guests.

Oct/Nov 2021

A security code is made up of 2 letters, then a 4-digit number. The letters come from $\{A, B, C, D, E\}$ and the digits come from $\{1, 2, 3, 4, 5, 6, 7\}$. No letter or digit can be used more than once. One example of such a code is BE3216.

Oct/Nov 2021

A Social Club has $15$ members, with $8$ men and $7$ women. Its committee is made up of $5$ members.

Oct/Nov 2022

Determine how many distinct arrangements of the 9 letters in the word ALLIGATOR have the two As adjacent and the two Ls adjacent.

Oct/Nov 2022

Determine how many distinct arrangements can be made from the $9$ letters in ACTIVATED.

Oct/Nov 2022

A restaurant has rectangular tables. Each table has space for four people: two seats are placed along each of the table’s longer sides (see diagram). Eight friends have reserved two tables, $X$ and $Y$. Rajid, Sue and Tan are three of these friends.

Oct/Nov 2023

Find the number of distinct arrangements of the 9 letters in the word ANDROMEDA in which no consonant is beside another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)

Oct/Nov 2023

Jai and his wife Kaz are hosting a party. Jai has invited five friends, and each friend will come with his wife.

Oct/Nov 2023

In how many distinct ways can the $9$ letters of INTELLECT be arranged if the two Ts are adjacent?

Oct/Nov 2024

Find how many different arrangements can be made from the 9 letters in the word ALGEBRAIC.

Oct/Nov 2024

Out of a class of $21$ students, $10$ are violinists, $6$ are guitarists and $5$ are pianists. A set of $7$ is to be selected from these $21$ students. The set will contain $4$ violinists, $2$ guitarists and $1$ pianist.

Oct/Nov 2024

Find how many distinct ways the $9$ letters in HAPPINESS can be arranged.

Oct/Nov 2024

In how many distinct ways can the 10 letters of the word SEYCHELLES be arranged?

Oct/Nov 2025

Pick 2 letters at random from the 10 letters in the word ZOOLOGICAL.

Oct/Nov 2025

The Splash Club has 26 members, including 16 swimmers and 10 divers. No person belongs to both the swimmer and diver groups. The committee is formed from 6 members chosen out of these 26.

Oct/Nov 2025

Find the number of distinct arrangements of the 9 letters in the word DAFFODILS where a D is placed at each end and the two Fs are not adjacent.

Oct/Nov 2025

Within a group of 25 athletes, the numbers are 8 sprinters, 5 hurdlers and 12 throwers.

Oct/Nov 2025