Given a set x with n elements, the number of subsets of x that have r elements is the **binomial coefficient** of n in r and is denoted as:

The expression above can also be written as C (n, r) or ^{n}C_{r.} The binomial coefficient indicates the number of ways in which an unordered subset of size r can be selected from a set of size n.

The binomial coefficient can be calculated using the following expression:

The formula above uses the factorial notation «!»; for any positive integer n, its factorial n! it is:

n! = n × (n - 1) × (n - 2)… × 3 × 2 × 1

Also, 0! = 1.

## P**properties **

^{n}C_{r}+^{n}C_{r + 1}=^{n + 1}C_{r + 1}^{n}C_{0 }+^{n}C_{1 }+^{n}C_{2 }+ ... +^{n}C_{n }= 2^{n}^{n}C_{0 }+^{n}C_{1 }+^{n}C_{2 }+ ... + (- 1)^{n}^{n}C_{n }= 0^{n}C_{0 }+^{n}C_{2 }+^{n}C_{4 }+^{n}C_{6 }+ ..._{ }= 2^{n-1}^{n}C_{0 }+^{n}C_{3 }+^{n}C_{5 }+^{n}C_{7 }+ ..._{ }= 2^{n-1}^{n}C_{n }–^{n + 1}C_{1 }+^{n + 2}C_{2 }-… +^{n + m}C_{n }=^{n + m + 1}C_{n + 1}_{ }^{n}C_{n }– (^{n}C_{1})^{2}+ (^{n}C_{2})^{2}_{ }- ... + (^{n}C_{n})^{2 }= 2^{n}C_{n }^{n}C_{1 }+ 2^{n}C_{2 }+ 3^{n}C_{3 }+… + N ·^{n}C_{n }= n · 2^{n-1}^{n}C_{0 }- two·^{n}C_{1 }+ 3^{n}C_{2 }+ ... + (-1)^{n + 1 }·^{ n}C_{n }= 0

**Example**: Determine the values of ^{10}C_{4} y ^{30}C_{18}.

We know that:

^{n}C_{r} = n! / [(n - r)! ∙ r! ]

Thus:

^{10}C_{4}= 10! / [(10 - 4)! ∙ 4! ] = 10! / [6! ∙ 4! ] = 210^{30}C_{18}= 30! / [(30 - 18)! ∙ 18! ] = 30! / [12! ∙ 18! ] = 8.6 x 10^{7}