Question

find the coefficient of x^3 y^3 in the expansion of (x+y)^n

Answer 1

Consider expansion of $(x + y)^n$:

$$(x + y)^n = \sum_{i=0}^{n} \binom{n}{i} x^i y^{n-i}$$

We need to find coefficient at term $x^3 y^3$.

We see that all terms of the expansion are in form $x^k y^{n-k}$. So if $n \neq 6$ coefficient at $x^3 y^3$ equals to 0.

If $n = 6$ the coefficient at $x^3 y^3$ equals to

$$\binom{6}{3} = \frac{6 \cdot 5 \cdot 4}{3!} = 20$$