Binomial Coefficient Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Binomial Coefficient.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

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

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The binomial coefficient $\binom{n}{k}$ counts how many ways to pick $k$ items from $n$ when order doesn't matter.

Common stuck point: The procedure for binomial coefficient is the easy part; the trap is using $\binom{n}{k}$ when order matters. Asking "Am I counting selections of $k$ from $n$ where order doesn't matter (or the matching expansion coefficient)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting selections of $k$ from $n$ where order doesn't matter (or the matching expansion coefficient)?

Worked Examples

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\}

Answer

\binom{6}{2} = 15

. Verified by listing all 15 two-item combinations.

First step

Apply formula:

\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15

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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

medium

Calculate

\binom{8}{3}

and explain what it counts.

Example 4

medium

Use Pascal's identity

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

to compute

\binom{6}{3}

from

\binom{5}{2}

and

\binom{5}{3}

Example 5

medium

Find the coefficient of

x^3

in the expansion of

(1+x)^7

Example 6

medium

In a binomial probability with

n=4

trials and

p=0.5

, find

P(X = 2)

where

X

counts successes.

Example 7

hard

Find the coefficient of

x^4

in the expansion of

(2+x)^7

Example 8

hard

(x+y)^6

, find the term containing

x^4 y^2

Example 9

hard

A fair coin is flipped

6

times. Find the probability of exactly

4

heads.

Example 10

challenge

Find

\sum_{k=0}^{10} \binom{10}{k}

and identify the closed form for any

n

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Calculate: (a)

\binom{4}{0}

, (b)

\binom{4}{4}

, (c)

\binom{4}{1}

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

easy

Compute

\binom{5}{2}

Example 4

easy

Compute

\binom{6}{0}

Example 5

easy

Compute

\binom{7}{7}

Example 6

easy

Compute

\binom{4}{1}

Example 7

easy

Use symmetry:

\binom{8}{6}

equals which simpler coefficient, and what is its value?

Example 8

easy

Compute

\binom{5}{3}

Example 9

easy

What is the coefficient of

x^2

in the expansion of

(1+x)^4

Example 10

easy

Why does

\binom{2}{5}

equal

0

(or is undefined as a count)?

Example 11

medium

Compute

\binom{10}{3}

Example 12

medium

A team of

4

is chosen from

9

players. How many possible teams?

Example 13

medium

Compute

\binom{6}{2} + \binom{6}{3}

and identify which single coefficient it equals via Pascal's rule.

Example 14

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 15

medium

Sum the row:

\binom{4}{0}+\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4}

. What is the total and why?

Example 16

medium

In how many ways can

3

identical prizes be given to

3

7

contestants (each at most one prize)?

Example 17

medium

A coin is flipped

5

times. How many outcome sequences have exactly

2

heads?

Example 18

medium

Compute

\binom{12}{2}

and

\binom{12}{10}

; explain why they are equal.

Example 19

medium

Compute

\binom{8}{3}

Example 20

challenge

Prove the symmetry identity

\binom{n}{k} = \binom{n}{n-k}

from the factorial formula.

Example 21

challenge

Prove Pascal's identity

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

by a counting argument.

Example 22

challenge

Show that the number of lattice paths from

(0,0)

(m,n)

using only unit right/up steps is

\binom{m+n}{m}

, and compute it for

(3,2)

Example 23

easy

Compute

\binom{6}{1}

Example 24

easy

Compute

\binom{10}{2}

Example 25

easy

Compute

\binom{9}{0} + \binom{9}{9}

Example 26

easy

Compute

\binom{7}{2}

Example 27

easy

Compute

\binom{12}{1}

Example 28

medium

A pizza shop offers

8

toppings. How many

3

-topping pizzas can you make if order doesn't matter and toppings are distinct?

Example 29

medium

How many

5

-card hands can be dealt from a standard

52

-card deck?

Example 30

medium

A team of

4

is chosen from

7

students. How many possible teams exist?

Example 31

medium

A code is a

4

-letter subset of

\{A,B,C,D,E,F,G,H\}

where order doesn't matter. How many such codes exist?

Example 32

hard

A class of

20

students must elect

5

to a committee, with one student (Alice) automatically included. How many committees are possible?

Example 33

hard

From

12

people, how many ways to form a committee of

5

that excludes a specific pair (Bob and Carol)?

Example 34

hard

From

15

books on a shelf, how many ways to choose a set of

5

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Related Concepts

Combination Factorial Binomial Theorem Binomial Distribution

Background Knowledge

These ideas may be useful before you work through the harder examples.

combinationfactorial