How to find $ \sum_{i=1}^{n} \sum_{j=1}^{n}(i+j) C_i C_j $ in the expansion of $(1+x)^n$?

Question

I am just a beginner to Binomial Theorem. I want to evaluate $$ \sum_{i=1}^{n} \sum_{j=1}^{n}(i+j) C_i C_j $$

Where $C_r$ are the binomial coefficients in the expansion of $(1+x)^n$. I don't know how to solve double summation.

I know that $(1+x)^n = ^{n}C_0 \ (1)^nx^0+ ^{n}C_1 \ 1^{n-1}x^1 + .... + ^{n}C_n1^0x^n$

Answer 1

Observe the variables of summation. If inner sum is over variable $j$ then it doesn't depend on the variable $i$, so you can take it common.

This leads to the following sum

$$2\left(\sum_{i=1}^{n} \binom{n}{i} \right)\cdot\left(\sum_{i=1}^{n}i \binom{n}{i} \right)$$

which can be easily handled.