What is $ {n\choose k}$?

Question

This is the Binomial theorem:

$$(a+b)^n=\sum_{k=0}^n{n\choose k}a^{n-k}b^k.$$

I do not understand the symbol $ {n\choose k}.$ How do I actually compute this? What does this notation mean? Help is appreciated.

Answer 1

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

It computes the number of ways we can choose $k$ items out of $n$ items.

Answer 2

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

With $n!=n\cdot (n-1)\cdot (n-2)\cdot\dotso\cdot 3\cdot 2\cdot 1=\prod_{i=1}^n i$

Answer 3

The symbol ${n\choose k}$ is read as "$n$ choose $k$." It represents the number of ways to choose $k$ objects from a set of $n$ objects. It has the following formula $$ {n\choose k}=\frac{n!}{(n-k)!k!}.$$ Here, $$ n!=n(n-1)(n-2)\cdots2\cdot1.$$

Answer 4

$${n\choose k}={n!\over k!(n-k)!}$$

Answer 5

Number of words of length $n$ over the alphabet $\{a,b\}$ such that $k$ letters are $a$. When you expand the LHS this characterization immediately implies the RHS.

Answer 6

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

It is used to calculate the number of ways "k" events can occur in "n" choices.