Permutations and Combinations: Mathematics 9709 (Cambridge International AS & A Level)

A tennis club has $9$ members who play singles. The club must fill three different roles (captain, vice-captain and social secretary) with three different members chosen from the $9$.

How many different ways can these three roles be filled?

Six friends (Priya, Quincy, Reyna, Soren, Tavita and Uche) stand in a row for a group photograph.

(a) Find the total number of different arrangements of the six friends in the row, with no restriction. [1]

(b) Reyna and Soren want to stand next to each other. Find the number of different arrangements in which Reyna and Soren stand next to each other. [3]

(c) Find the number of different arrangements in which Reyna and Soren do not stand next to each other. [2]

(d) Find the number of different arrangements in which Tavita stands at one of the two ends of the row (with no restriction on the other five friends). [2]

Consider the $10$ letters of the word BOOKKEEPER.

(a) Find the number of different arrangements of these $10$ letters. [1]

(b) Find the number of different arrangements of these $10$ letters in which the two Ks are next to each other. [4]

(c) Find the number of different arrangements of these $10$ letters in which the two Ks are not next to each other. [2]

A university robotics lab has $7$ senior researchers and $5$ junior researchers. A demonstration team of $6$ people is to be selected from these $12$ researchers.

(a) Find the number of different teams that can be selected if there is no restriction on the number of senior or junior researchers chosen. [1]

(b) Find the number of different teams that can be selected if the team must include exactly $2$ junior researchers (and therefore $4$ senior researchers). [3]

(c) Find the number of different teams that can be selected if the team must include at least $4$ senior researchers. [4]

A school debate club has $10$ members: $6$ boys and $4$ girls. A delegation of $4$ members is chosen at random to attend a competition.

How many different delegations of $4$ members include at least one girl?

A café offers a meal deal in which a customer chooses exactly one starter from $4$ different starters, exactly one main course from $6$ different main courses, and exactly one dessert from $3$ different desserts.

How many different meal deals can a customer choose?

A quiz team of $5$ students is to be chosen from a group of $12$ students. Two of the students, Farah and Idris, are on rival debating teams and refuse to be selected for the quiz team together.

(a) Find the number of different quiz teams that can be chosen if there is no restriction. [1]

(b) Find the number of different quiz teams that include both Farah and Idris. [3]

(c) Find the number of different quiz teams in which Farah and Idris are not both included. [2]

A $4$-digit security code is formed by arranging $4$ of the $10$ digits $0,1,2,\ldots,9$ in a row, with no digit used more than once.

(a) Find the number of different $4$-digit sequences of distinct digits that can be formed, with no other restriction. [1]

(b) A valid code cannot start with the digit $0$. Find the number of different valid codes. [3]

(c) Find the number of different valid codes (still with distinct digits and not starting with $0$) that are even, that is, the last digit is $0$, $2$, $4$, $6$ or $8$. [4]

Consider the $9$ letters of the word ADDRESSES.

(a) Find the number of different arrangements of these $9$ letters. [1]

(b) Find the number of different arrangements of these $9$ letters in which the three Ss are next to each other. [3]

(c) Find the number of different arrangements of these $9$ letters that start with the letter A. [2]

The digits $1$, $2$, $3$, $4$, $5$ are each used at most once to form a $2$-digit number, so the two digits used must be different from each other.

How many of these $2$-digit numbers are odd?