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) counts the number of ways to choose k items from n distinct items without regard to order. It equals 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) to compute (63) from (52) and (53).

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−k, find P(X=3) (exactly 3 heads).

Example 4

easy
True or false: (nk) counts ordered arrangements.

Example 5

easy
Compute (52).

Example 6

easy
How many ways can you choose 3 books from a shelf of 5?

Example 7

medium
Compute (104).

Example 8

easy
Use symmetry: (86) equals which simpler coefficient, and what is its value?

Example 9

medium
Compute (83).

Example 10

medium
A team of 4 is chosen from 7 students. How many possible teams exist?

Example 11

easy
Compute (121).

Example 12

medium
How many ways to choose a 5-card hand from a 52-card deck? Express as a binomial coefficient and give its value.

Example 13

medium
In how many ways can 3 identical prizes be given to 3 of 7 contestants (each at most one prize)?

Example 14

hard
In (x+y)6, find the term containing x4y2.

Example 15

easy
Compute (77).

Example 16

easy
Compute (72).

Example 17

challenge
Find ∑k=010(10k) and identify the closed form for any n.

Example 18

easy
Compute (53).

Example 19

hard
Find the coefficient of x4 in the expansion of (2+x)7.

Example 20

easy
Compute (60).
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