Expectation of uniform random variable knowing sum of $n$ identical uniform variables.

Question

Let $X_1, ..., X_n$ be independent and identically distributed random variables on $[0,1]$. Find: $$ \mathbb{E}[X_1|X_1 + ... + X_n = x] $$

Answer 1

Hint: for each $k$ $$ \mathbb{E}[X_1|X_1 + ... + X_n = x] = \mathbb{E}[X_k|X_1 + ... + X_n = x] $$

Now what happens if you make the sum?