Practice Binomial Coefficient in Math

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

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Quick Recap

The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn distinct items without regard to order. It equals n!k!(n−k)!\frac{n!}{k!(n-k)!}.

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

Showing a random 20 of 50 problems.

Example 1

medium
Use Pascal's identity (nk)=(n−1k−1)+(n−1k)\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k} to compute (63)\binom{6}{3} from (52)\binom{5}{2} and (53)\binom{5}{3}.

Example 2

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?

Example 3

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

Example 4

easy
True or false: (nk)\binom{n}{k} counts ordered arrangements.

Example 5

easy
Compute (52)\binom{5}{2}.

Example 6

easy
How many ways can you choose 33 books from a shelf of 55?

Example 7

medium
Compute (104)\binom{10}{4}.

Example 8

easy
Use symmetry: (86)\binom{8}{6} equals which simpler coefficient, and what is its value?

Example 9

medium
Compute (83)\binom{8}{3}.

Example 10

medium
A team of 44 is chosen from 77 students. How many possible teams exist?

Example 11

easy
Compute (121)\binom{12}{1}.

Example 12

medium
How many ways to choose a 55-card hand from a 5252-card deck? Express as a binomial coefficient and give its value.

Example 13

medium
In how many ways can 33 identical prizes be given to 33 of 77 contestants (each at most one prize)?

Example 14

hard
In (x+y)6(x+y)^6, find the term containing x4y2x^4 y^2.

Example 15

easy
Compute (77)\binom{7}{7}.

Example 16

easy
Compute (72)\binom{7}{2}.

Example 17

challenge
Find ∑k=010(10k)\sum_{k=0}^{10} \binom{10}{k} and identify the closed form for any nn.

Example 18

easy
Compute (53)\binom{5}{3}.

Example 19

hard
Find the coefficient of x4x^4 in the expansion of (2+x)7(2+x)^7.

Example 20

easy
Compute (60)\binom{6}{0}.
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