Practice Binomial Coefficient in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The number of ways to choose k items from n items, written C(n, k) or \binom{n}{k}.

Same as combination count, but now viewed as a coefficient in algebraic expansions.

Example 1

medium
Calculate \binom{6}{2} using the formula \binom{n}{k} = \frac{n!}{k!(n-k)!}, and verify by listing all combinations of 2 items from \{A, B, C, D, E, F\}.

Example 2

hard
A fair coin is flipped 5 times. Using P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, find P(X=3) (exactly 3 heads).

Example 3

easy
Calculate: (a) \binom{4}{0}, (b) \binom{4}{4}, (c) \binom{4}{1}.

Example 4

hard
A committee of 3 is chosen from 8 people. How many possible committees exist? If one specific pair (Alice and Bob) must both be included, how many committees include both?
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