15 Combinations of 5

Evaluate the combination:
₁₅C₅

Combination Definition:

A unique order or arrangement

Combination Formula:

_nC_r =	n!
	r!(n - r)!

where n is the number of items
r is the unique arrangements.

Plug in n = 15 and r = 5

₁₅C₅ 2	15!
	5!(15 - 5)!

Factorial Formula:

n! = n * (n - 1) * (n - 2) * .... * 2 * 1

Calculate the numerator n!:

n! = 15!
15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
15! = 1,307,674,368,000

Calculate (n - r)!:

(n - r)! = (15 - 5)!
(15 - 5)! = 10!
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
10! = 3,628,800

Calculate r!:

r! = 5!
5! = 5 x 4 x 3 x 2 x 1
5! = 120

Calculate ₁₅C₅

₁₅C₅ =	1,307,674,368,000
	120 x 3,628,800

₁₅C₅ =	1,307,674,368,000
	435,456,000

Excel or Google Sheets formula:

=COMBIN(15,5)

What is the Answer?

₁₅C₅ = 3,003

How does the Permutations and Combinations Calculator work?

Free Permutations and Combinations Calculator - Calculates the following:
Number of permutation(s) of n items arranged in r ways = _nP_r
Number of combination(s) of n items arranged in r unique ways = _nC_r including subsets of sets
This calculator has 2 inputs.

What 2 formulas are used for the Permutations and Combinations Calculator?

nPr=n!/r!
nCr=n!/r!(n-r)!

For more math formulas, check out our Formula Dossier

What 4 concepts are covered in the Permutations and Combinations Calculator?

combination: a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter
_nP_r = n!/r!(n - r)!
factorial: The product of an integer and all the integers below it
permutation: a way in which a set or number of things can be ordered or arranged.
_nP_r = n!/(n - r)!
permutations and combinations

Example calculations for the Permutations and Combinations Calculator

Permutations and Combinations Calculator Video

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