Prove that $\binom{n+2} {k} = \binom{n} k + 2\binom n {k-1} + \binom n {k-2}$

Question

I am trying to solve the following combinatorics question:

Prove that

$$\binom{n+2} {k} = \binom{n} k + 2\binom n {k-1} + \binom n {k-2}$$

Answer 1

Hint:

$$\binom{n+2}{k}=\binom{n+1}{k}+\binom{n+1}{k-1}\tag{1}$$Can you take it from here? You can use relation $(1)$ more than once, for instance how can you express: $$\binom{n+1}{k}=\dots\\\binom{n+1}{k-1}=\dots$$

Answer 2

We have $n+2$ balls, $2$ are red and $n$ blue (all of different sizes).

On how many way can we choose $k$ balls?

Answer:

We can chose $0$ red and $k$ blue balls, that is ${2\choose 0}{n\choose k}$.

We can chose $1$ red and $k-1$ blue balls, that is ${2\choose 1}{n\choose k-1}$ combinations.

We can chose $2$ red and $k-2$ blue balls, that is ${2\choose 2}{n\choose k-2}$ combinations.