Quick questions on Combinations: counting unordered selections with $\binom{n}{r}$

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is complementary counting?

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For "at least

k

" or "at most

k

" problems it is usually far quicker to count the complement than to add up every allowed case:

What are splitting into groups?

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To split

n

distinct objects into groups of sizes

r_1, r_2, \dots, r_k

(with

r_1 + \dots + r_k = n

What is symmetry the wrong way?

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\binom{20}{17} = \binom{20}{3}

because

20 - 17 = 3

. Subtract from the top number; do not just swap digits.

What is pascal's rule sign?

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\binom{n}{r} = \binom{n - 1}{r - 1} + \binom{n - 1}{r}

has a plus. There is no subtraction.

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