Combination Examples in Math

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

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

Concept Recap

A combination is an unordered selection of objects — the number of ways to choose $r$ items from $n$ distinct items is $C(n,r) = \frac{n!}{r!(n-r)!}$ .

How many ways to choose a group? $\{A, B, C\} = \{C, B, A\}$ .

Read the full concept explanation →

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: A combination counts the ways to pick $r$ items from $n$ when rearranging the picked items doesn't make a new selection.

Common stuck point: The procedure for combination is the easy part; the trap is forgetting to divide by $r!$ . Asking "Does rearranging the chosen items leave it the same selection?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does rearranging the chosen items leave it the same selection?

Worked Examples

Example 1

easy

A committee of

3

is to be chosen from

8

people. How many different committees are possible?

Answer

\binom{8}{3} = 56

First step

Recall the combination formula for unordered selections:

\binom{n}{r} = \frac{n!}{r!(n-r)!}

, with

n = 8

r = 3

Full solution

2
Cancel common factorial terms: $\binom{8}{3} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}$
3
Calculate: $\frac{336}{6} = 56$

Combinations count selections where order does not matter. Choosing members A, B, C for a committee is the same as choosing C, B, A.

Example 2

medium

From a group of

6

men and

4

women, how many committees of

5

can be formed that include exactly

3

men and

2

women?

Example 3

easy

A jar holds 5 different fruits. How many distinct fruit salads of 3 different fruits can be made? (Order does not matter inside a salad.)

Example 4

medium

From 10 marbles (4 red, 6 blue), how many ways to pick 3 marbles with exactly 1 red?

Example 5

medium

From 8 people, choose a group of 4 in which Alice is always included.

Example 6

hard

Solve

C(n,3)=10\cdot C(n,1)

for

n

Example 7

hard

A 4-card hand is drawn from a 52-card deck. What is the probability the hand contains all 4 aces?

Example 8

challenge

Use Pascal's identity to compute

C(7,3)

from

C(6,2)

and

C(6,3)

Practice Problems

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

Example 1

medium

How many ways can you choose

4

books from a shelf of

10

books?

Example 2

medium

A pizza shop offers

8

toppings. How many different

3

-topping pizzas can be made if order does not matter?

Example 3

easy

Compute

C(5,2)

Example 4

easy

Compute

C(6,6)

Example 5

easy

How many ways to choose 2 toppings from 5 (order does not matter)?

Example 6

easy

Compute

C(7,0)

Example 7

easy

Compute

C(8,1)

Example 8

easy

Use symmetry to find

C(10,8)

Example 9

easy

How many handshakes occur if 4 people each shake hands once with every other?

Example 10

easy

Compute

C(4,2)

Example 11

medium

A committee of 3 is chosen from 7 people. How many committees are possible?

Example 12

medium

From 5 men and 4 women, how many ways to choose a group of 2 men and 2 women?

Example 13

medium

Compute

C(8,3)

Example 14

medium

How many ways to choose 3 cards from a standard 52-card deck (order irrelevant)?

Example 15

medium

Solve for

n

C(n,2)=15

Example 16

medium

How many diagonals does a convex hexagon (6 vertices) have?

Example 17

medium

How many ways to split 6 distinct items into a chosen group of 4 and leave 2 behind?

Example 18

medium

From 10 students, how many ways to pick a team of 3 if 2 specific students must both be on it?

Example 19

medium

Compute

C(9,2)

Example 20

challenge

How many 5-card hands from 52 cards contain exactly 2 aces?

Example 21

challenge

Prove and use:

C(n,r)+C(n,r-1)=C(n+1,r)

to compute

C(5,2)+C(5,1)

Example 22

challenge

In how many ways can 8 identical balls be placed into 3 distinct boxes (any box may be empty)?

Example 23

easy

Compute

C(6,2)

Example 24

easy

How many ways can a teacher choose 2 students from a class of 9 to be hall monitors?

Example 25

easy

Compute

C(11,1)

Example 26

easy

How many ways to choose a 4-person subcommittee from a group of 6?

Example 27

medium

From 7 boys and 5 girls, how many ways to form a committee of 4 with exactly 2 boys and 2 girls?

Example 28

medium

Solve for

n

C(n,2)=21

Example 29

medium

A convex octagon (8 vertices) has how many diagonals?

Example 30

medium

Compute

C(10,3)+C(10,7)

Example 31

medium

A pizza shop offers 10 toppings. How many distinct pizzas use 4 different toppings?

Example 32

medium

From 8 people, choose 4 such that two specific people, Bob and Carol, are NOT both included.

Example 33

medium

How many ways to choose 3 different letters from the word LEARN?

Example 34

hard

A 5-card hand is dealt from a standard deck. How many hands contain exactly 3 hearts?

Example 35

hard

How many subsets of

\{1,2,\ldots,10\}

have exactly 4 elements and contain the number 1?

Example 36

hard

How many ways to distribute 10 identical candies among 4 children so that each child gets at least 1?

Example 37

hard

From 12 students, how many ways to form a 5-person team that includes at least 1 of the 3 captains?

Example 38

challenge

Find the coefficient of

x^4

in the expansion of

(1+x)^{10}

Example 39

challenge

How many lattice paths go from

(0,0)

(5,4)

using only unit steps right or up?

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Related Concepts

Permutation Factorial Binomial Coefficient

Background Knowledge

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

permutationfactorial