AS Level Mathematics · Probability & Statistics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
This topic is about counting without listing. A permutation is an arrangement, where order matters. A combination is a selection, where order does not matter. First decide which one the question needs: a line of people or a number made from digits is an arrangement; a committee or a team is a selection.
When letters or objects are repeated, swapping identical ones gives nothing new, so divide by the factorial of each repeat.
With a restriction, deal with it first. Items that must be together are treated as one block. For items that must not be together, find the total and subtract the cases where they are together, or place them in the gaps.
Using permutations for a selection such as a committee, which counts each group many times. If changing the order does not give a new answer, use combinations.
Answer: 360
Try each one, then open the answer.
Answer: B. There are 3 vowels (O, A, E). The first and last letters can be chosen in 3 × 2 = 6 ways and the other 5 letters arranged in 5! = 120 ways: 6 × 120 = 720. Using C(3, 2) = 3 for the ends ignores their order and gives 360.
Answer: D. Treat the pair as one unit: 4 units can be arranged in 4! = 24 ways, and the pair can be ordered in 2 ways, giving 48. There are 120 − 48 = 72 arrangements with the pair apart.
Answer: D. Treat the girls as one block: 5 units arranged in 5! = 120 ways, and the girls can be ordered within the block in 3! = 6 ways. Total 120 × 6 = 720.
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