Combination (Math)
See how to solve a combination (nCr).
4 examples and their solutions.
Combination
Formula
r numbers nCr = n⋅(n - 1)⋅(n - 2)⋅ ...r!
Meaning: From n things, pick r things.
(no arrangement)
Permutation
Factorial
nCr = nCn - r - [1]
nC1 = n - [2]
nC0 = nCn = 1 - [3]
nC1 = n - [2]
nC0 = nCn = 1 - [3]
[1]
From n things,
(number of ways to pick r things)
= (number of ways to leave (n - r) things)
(number of ways to pick r things)
= (number of ways to leave (n - r) things)
[2]
From n things, pick 1 thing
→ n ways
→ n ways
[3]
From n things,
(number of ways to pick 0 things)
= (number of ways to leave n things)
→ Don't do anything.
→ 1 way
(number of ways to pick 0 things)
= (number of ways to leave n things)
→ Don't do anything.
→ 1 way
Example
7C3
Solution 7C3
= 7⋅6⋅53!
= 7⋅6⋅53⋅2⋅1
= 7⋅5
= 35
= 7⋅6⋅53!
= 7⋅6⋅53⋅2⋅1
= 7⋅5
= 35
Close
Example
8C6
Solution 8C6
= 8C2 - [1]
= 8⋅72!
= 8⋅72⋅1
= 4⋅7
= 28
= 8C2 - [1]
= 8⋅72!
= 8⋅72⋅1
= 4⋅7
= 28
[1]
8C6 = 8C8 - 6
= 8C2
= 8C2
Close
Example
5 cups, 6 spoons
Find the number of ways to choose 2 cups and 3 spoons.
Solution Find the number of ways to choose 2 cups and 3 spoons.
N = 5C2⋅6C3 - [1]
= 5⋅42!⋅6⋅5⋅43!
= 5⋅42⋅1⋅6⋅5⋅43⋅2⋅1
= 5⋅2 ⋅ 5⋅4
= 10⋅20
= 200
= 5⋅42!⋅6⋅5⋅43!
= 5⋅42⋅1⋅6⋅5⋅43⋅2⋅1
= 5⋅2 ⋅ 5⋅4
= 10⋅20
= 200
[1]
From 5 cups,
pick 2 cups.
→ 5C2
From 6 spoons,
pick 3 spoons.
→ 6C3
Picking a cup and picking a spoon don't affect each other.
→ Multiply these two.
Number of Ways (Math)
pick 2 cups.
→ 5C2
From 6 spoons,
pick 3 spoons.
→ 6C3
Picking a cup and picking a spoon don't affect each other.
→ Multiply these two.
Number of Ways (Math)
Close
Example
3 letters: a, b, c
4 numbers: 1, 2, 3, 4
Find the number of ways to choose 2 letters and 2 numbers
and arrange them in a row.
Solution 4 numbers: 1, 2, 3, 4
Find the number of ways to choose 2 letters and 2 numbers
and arrange them in a row.
N = 3C2⋅4C2⋅4! - [1] [2]
= 3C1⋅4C2⋅4!
= 3⋅4⋅32!⋅4!
= 3⋅4⋅32⋅1⋅4⋅3⋅2⋅1
= 3⋅2⋅3⋅4⋅3⋅2
= 6⋅12⋅6
= 72⋅6
= 432
= 3C1⋅4C2⋅4!
= 3⋅4⋅32!⋅4!
= 3⋅4⋅32⋅1⋅4⋅3⋅2⋅1
= 3⋅2⋅3⋅4⋅3⋅2
= 6⋅12⋅6
= 72⋅6
= 432
[1]
From 3 letters,
pick 2 letters.
→ 3C2
From 4 numbers,
pick 2 numbers.
→ 4C2
Picking letters and picking numbers don't affect each other.
→ Multiply these two.
pick 2 letters.
→ 3C2
From 4 numbers,
pick 2 numbers.
→ 4C2
Picking letters and picking numbers don't affect each other.
→ Multiply these two.
[2]
Close