Permutations and Combinations

Revise Permutations and Combinations for Probability & Statistics 1 S1 (AS Level) — revision notes and instant AI marking.

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Cambridge International AS & A Level Mathematics 9709 — Probability & Statistics 1

Permutations and Combinations

The big idea: Ask one question first: does order matter? If it does, you are arranging (permutations); if not, you are choosing (combinations).

Summary — What This Chapter Covers

n distinct objects can be arranged in a line in n! ways.
With repeated objects, divide by the factorial of each repeat count.
Objects that must be together: glue them into one unit, then arrange inside the unit.
Objects that must be apart: total arrangements − arrangements with them together (or slot them into gaps).
Selections where order does not matter: ⁿC_r = n!/(r!(n − r)!). Arrangements of r from n: ⁿP_r = n!/(n − r)!.
Circular arrangements are not examined.

1. Arrangements

Permutations

Lines, repeats and restrictions

Arranging n different objects in a line: n choices for the first place, n − 1 for the second, and so on, giving n!. If some objects are identical, swapping them changes nothing, so divide by the number of ways of arranging each identical group.

Together: treat the group as a single block, arrange the blocks, then multiply by the arrangements inside the block. Not together: subtract the "together" count from the total, or arrange the others first and place the restricted objects in the gaps. For seats in rows, place the restricted people first, then arrange everyone else in the remaining seats.

Arrangements n! With repeats: n!/(p! q! …) r from n in order: ⁿP_r = n!/(n − r)!

Worked Example 1

Find the number of arrangements of the letters of NEEDLESS, and the number in which the two S's are together.

Worked Example 2

Six people stand in a line. In how many ways can this be done if two particular people must stand next to each other? If they must not?

Practice Question 1

Seven people sit in two rows: 3 in the front, 4 in the back. Two particular people must both sit in the front row. How many arrangements are there?

2. Selections

Combinations

Order does not matter

Choosing r objects from n distinct objects without regard to order gives ⁿC_r selections. For "at least" conditions, either add the separate cases or subtract the unwanted cases from the total. For selections from separate groups (men and women, red and blue), multiply the choices from each group within a case, then add the cases.

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