9709 · 5.2
Permutations and combinations flashcards
Revision flashcards for Cambridge 9709 Permutations and combinations (syllabus 5.2). Flip, recall, then mark a real past-paper question.
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What is a permutation?
An arrangement of a set of objects in a specific order. Use when the order of selection is important.
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What is the formula for permutations?
$^n P_r = \frac{n!}{(n-r)!}$, where $n$ is the total number of items and $r$ is the number of items to arrange.
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What is a combination?
A selection of a set of objects where the order of selection does not matter. Use when forming groups or committees.
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What is the formula for combinations?
$^n C_r = \binom{n}{r} = \frac{n!}{(n-r)!r!}$, where $n$ is the total number of items and $r$ is the number of items to choose.
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What is the value of $0!$?
$0! = 1$. This is a mathematical convention that ensures formulas like $^n P_n = n!$ remain consistent.
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How do you arrange $n$ distinct objects?
There are $n!$ ways to arrange $n$ distinct objects. For example, the letters in 'ABCDE' can be arranged in $5! = 120$ ways.
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How do you arrange items with repetitions, like in the word 'STATISTICS'?
Use the formula $\frac{n!}{p!q!...}$. For 'STATISTICS', there are 10 letters total, with 3 'S', 3 'T', and 2 'I'. The number of arrangements is $\frac{10!}{3!3!2!}$.
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What is the relationship between $^n P_r$ and $^n C_r$?
$^n P_r = ^n C_r \times r!$. An arrangement ($P$) is a selection ($C$) followed by arranging the selected items ($r!$ ways).
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How do you handle a condition where certain items must be kept together?
Treat the items that must be together as a single 'block'. Arrange the blocks and the other items. Then, multiply by the number of ways to arrange the items within the block.
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What is a common strategy for 'at least one' problems?
Calculate the total number of possibilities without any restrictions, and subtract the number of ways the unwanted scenario can happen. For example, 'at least one man' = (Total selections) - (selections with zero men).
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When do you add possibilities and when do you multiply?
Add when you have mutually exclusive cases (e.g., choosing a committee of 3 men OR 3 women). Multiply when events happen in sequence or together (e.g., choosing 2 men AND 2 women).