Permutations and Combinations A/AS level part 1

- September 19, 2024

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics used to count the number of ways to arrange or select items. The choice between permutations and combinations depends on whether the order of the items matters.

Permutations

Permutations are used when the order of items matters. The formula for permutations of n items taken r at a time is:

$$P(n, r) = \frac{n!}{(n-r)!}$$

Where n! (n factorial) is the product of all positive integers up to n.

Combinations

Combinations are used when the order of items does not matter. The formula for combinations of n items taken r at a time is:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Example Applications

(1) Selecting Teams or Committees

Problem: How many ways can a team be chosen from a group?
Solution: Use combinations when the order does not matter.
Formula: $$C(n, r)$$ where n is the total number of people and r is the number of people to choose.

(2) Arranging Letters or Objects

Problem: How many different ways can letters be arranged?
Solution: Use permutations when the order matters.
Formula: If there are repeated letters, divide by the factorial of the number of repetitions for each letter.
Example: For the word "DECEIVED", with repeated E's and D's, use $$\frac{8!}{3!2!}$$.

(3) Constraints in Arrangements or Selections

Problem: How many arrangements meet specific conditions (e.g., certain elements must be together)?
Solution: Treat groups that must stay together as single units, then arrange these units.
Example: To arrange "DECEIVED" with all E's together, treat "EEE" as one unit.

These principles and formulas provide a framework for solving various combinatorial problems involving selections and arrangements.

Exercise

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

(a) In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
(b) 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.

(c) How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?

80. 9709_s22_qp_51

(a) 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.

(b) Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.

81. 9709_s22_qp_51

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.

(a) In how many different ways can the committee be selected if exactly one of Mr Lan and Mrs Lan must be on the committee?
(b) 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?

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(a) Find the number of different arrangements of the 9 letters in the word CROCODILE.

(b) Find the number of different arrangements of the 9 letters in the word CROCODILE in which there is a C at each end and the two Os are not together.

(c) Four letters are selected from the 9 letters in the word CROCODILE. Find the number of selections in which the number of Cs is not the same as the number of Os.

(d) Find the number of ways in which the 9 letters in the word CROCODILE can be divided into three groups, each containing three letters, if the two Cs must be in different groups.

83. 9709_s22_qp_53

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.

(a) In how many different ways can the remaining 12 members of the group be divided between the three cars?
(b) In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?
(c) In how many ways can the team be chosen from the group so that all three children are from different families?
(d) In how many ways can this team be chosen so that at least one of Mr Kenny or Mr Lizo is included?

84. 9709_m21_qp_52

(a) Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.

(b) Find the total number of different arrangements of these letters where there is an R at both ends, and two As are not together.

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