# Binomial and multinomial coefficients¶

The binomial coefficient $$\binom n k$$ represents the number of different ways to choose a subset of size $$k$$ out of a set of size $$n$$, and is in general given by the formula

$$\displaystyle \binom n k = \frac{n!}{k!(n-k)!}$$

However, binomial coefficients can be computed more efficiently than literally using the above formula with factorial, so Disco has special built-in support for computing them. Since $$\binom n k$$ is usually pronounced “$$n$$ choose $$k$$”, the Disco syntax is n choose k. For example:

Disco> 5 choose 2
10
Disco> 7 choose 0
1
Disco> 0 choose 0
1
Disco> 7 choose 8
0
Disco> 100 choose 23
24865270306254660391200


## Multinomial coefficients¶

Disco also has support for multinomial coefficients:

$$\displaystyle \binom{n}{k_1 \quad k_2 \quad \dots \quad k_r} = \frac{n!}{k_1! k_2! \dots k_r! (n - k_1 - k_2 - \dots - k_r)!}$$

is the number of ways to simultaneously choose subsets of size $$k_1, k_2, \dots, k_r$$ out of a set of size $$n$$. In Disco, a multinomial coefficient results when the second argument to choose is a list instead of a natural number. For example:

Disco> 10 choose 2
45
Disco> 10 choose 
45
Disco> 10 choose [2,3]
2520
Disco> 10 choose [2,3,5]
2520
Disco> 10 choose [2,3,5] == (10 choose 2) * (8 choose 3) * (5 choose 5)
true