Combination & Permutation Calculator (nCr & nPr)

Combinations and Permutations Explained

In combinatorial mathematics, counting problems arise whenever we need to determine the number of possible outcomes, groupings, or sequences from a finite collection of items.

The Four Combinatorial Archetypes

Depending on whether order matters and whether repetition is allowed, we have four fundamental formulas:

Type	Order Matters?	Repetition?	Formula	Notation
Combination	No	No	$\frac{n!}{r!(n - r)!}$	$C(n, r) \text{ or } \binom{n}{r}$
Permutation	Yes	No	$\frac{n!}{(n - r)!}$	$P(n, r) \text{ or } ^n P_r$
Comb. with Repetition	No	Yes	$\frac{(n + r - 1)!}{r!(n - 1)!}$	$C(n + r - 1, r)$
Perm. with Repetition	Yes	Yes	$n^r$	$n^r$

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Step-by-Step Factorial Simplification

Consider calculating $\binom{8}{3}$ (choosing 3 members from an 8-person committee):

$$\binom{8}{3} = \frac{8!}{3!(8 - 3)!} = \frac{8!}{3! \times 5!}$$

Notice that the $5!$ in the denominator cancels out the lower factors of $8!$:

$$\binom{8}{3} = \frac{8 \times 7 \times 6 \times \cancel{5!}}{(3 \times 2 \times 1) \times \cancel{5!}} = \frac{8 \times 7 \times 6}{6} = 8 \times 7 = \mathbf{56}$$

By simplifying before multiplying, calculations remain efficient and mathematically exact.