Number of Combinations

Topics: Combination - Combinatorics

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When combining $k$ elements from a set of $n$ elements (i.e. forming $k$ combinations, $k\le n$), the number of combinations we can have is:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$$

Permutations divided by $k!$

Notice that this is basically the formula for permutations with no repetitions, but divided by $k!$.

This is because each $k$-combination of a set of $n$ elements has $k!$ permutations, and when combining, we do not care about the order of the elements.

Properties §

(theorem)

When $k>n$, then:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=0$$

When $k=n$, then:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=1$$

When $k=0$, then:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=1$$

When $k=1$, then:

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=n$$