body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Rule

Combination Formula

The number of combinations of n different objects taken r at a time — denoted as _nC_r — is given by the following formula.

_nC_r=n!/r!(n-r)!, r≤ n

The exclamation mark in the formula indicates that the factorial of the value should be calculated. As a direct consequence of the above formula, since 0!=1, when n=r the number of combinations is 1.

_nC_n=1

An alternative notation for _cC_r is C(n,r).

Proof

The formula can be proven by using the Permutation Formulas. _nP_r=n!/(n-r)! and _rP_r=r! Let _nC_r be the number of combinations of n objects chosen r at a time. By the Fundamental Counting Principle, the product of _nC_r by _rP_r equals the number of permutations of r objects out of n. _nC_r* _rP_r = _nP_r ⇓ _nC_r* r!= n!/(n-r)! Finally, by applying the Division Property of Equality, the Combination Formula is obtained.

_nC_r* r!= n!/(n-r)! ⇕ _nC_r=n!/r!(n-r)!

Exercises

Recommended exercises

To get personalized recommended exercises, please login first.

Return to lesson.

Edit Lesson