Permutations and Combinations Worksheet

This below worksheet containing the solved example shows how to compute Permutation & Combination.

Permutation
The number of arrangements that can be made out of n things taking r at a time is called the number of permutations of n things taken r at a time.
1. npr = n (n - 1) (n - 2) .... (n - (r - 1))
2. npr = n! / (n - r)!
3. npn = n!
4. np0 = 1
5. The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second kind such that p + q = n is n! /p!q!
6. The number of circular permutations of n distinct objects is (n - 1)!
7. If there are n things and if the direction is not taken into consideration, the number of circular permutation is (n - 1)! /2

Combination
A selection of any r things out of n things is called Combination of n things r at a time.
1. nCr = n! /r!(n - r)!
2. nCr = nPr /r!
3. nCn = 1
4. nC0 = 1
5. nCr = nCn - r
6. If x and y are non-negative integers such that x + y = n then nCx = nCy .
7. If n and r are positive integers such that r <= n, then nCr + nCn - r = n + 1Cr .
8. If n and r are positive integers such that 1 <= r <= n, then nCr = (n / r) (n - 1)C(r - 1)
9. For any positive integers x and y nCx = nCy => x = y (or) x + y = n.

Difference between Permutation and combination
1. In a combination only selection is made whereas in a permutation not only a selection is made but also as arrangement in a definite order is considered.
2. Usually the number of permutation exceeds the number of combinations.
3. Each combination corresponds to many permutations.

When you try yourself such calculations, this Permutation and Combination Calculator can be used to verify your results.