Binomial Coefficient¶

Definition¶

Let $n,k\in\mathbb{N}$ with $0\le k\le n$ $n$ . The binomial coefficient $\binom nk$ is the number of ways to form a combination of $k$ objects from $n$ distinct objects without regard to order, and is given by

\binom nk=\frac{n!}{k!(n-k)!}.

Interpretation¶

The binomial coefficient $\binom nk$ , read " $n$ choose $k$ ", counts the subsets with $k$ elements of a set with $n$ elements: the number of ways to choose $k$ objects when only which objects are chosen matters, not their order.

Pascal's triangle for n = 0 to 5, with the entry 10 in row 5 highlighted as the sum of the two entries 4 and 6 above it.

Row $n$ of Pascal's triangle lists $\binom n0,\binom n1,\ldots,\binom nn$ from left to right. The highlighted entry is $\binom52=\binom41+\binom42=4+6=10$ $10$ , and this sum has a counting reason: fix one of the five objects; a pair either contains it, and then one partner is chosen from the other four, or it does not, and then both are chosen from the other four. Each row also sums to $2^n$ , here $1+5+10+10+5+1=32$ $32$ , because every element is either in a subset or not, which gives $2^n$ subsets of all sizes together.

The formula reads as "order, then forget the order": $n!/(n-k)!$ counts the ordered selections, the $k$ -permutations, and each unordered group of $k$ objects appears among them $k!$ times, so dividing by $k!$ leaves the number of combinations. The symmetry $\binom nk=\binom n{n-k}$ is the left-right symmetry of each row.

A common slip is to stop at $n!/k!$ or at $n!/(n-k)!$ $k)!$ ; only the full formula counts unordered groups. Binomial coefficients are the basic tool of naive probability when the outcomes are hands, committees or samples, and they owe their name to the expansion of $(a+b)^n$ in the Key Results.

Key Results¶

$\binom nk=\binom n{n-k}$ $(n - k n)$ , since choosing $k$ objects is equivalent to leaving out $n-k$ objects.
$\binom n0=\binom nn=1$ and $\binom n1=n$ $n$ .
Binomial coefficients use factorials and appear in the expansion $\displaystyle (a+b)^n=\sum_{k=0}^n\binom nk a^k b^{n-k}$ $k = 0 \sum n (k n) a^{k} b^{n - k}$ .

Examples¶

From $10$ applicants, the number of possible interview groups of size $3$ is $\binom{10}{3}$ .
The number of ways to choose $5$ cards from a $52$ card deck is $\binom{52}{5}$ .
In a binomial distribution, $\binom nk$ counts the number of ways to choose which $k$ of the $n$ trials are successes, which is why it appears in $\mathbb{P}(X=k)=\binom nk p^k(1-p)^{n-k}$ $p)^{n - k}$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Aliases: n choose k, choose function Keywords: combination number Symbols: element of, natural numbers, sum

Last updated: 20 October 2026 03:02