## Task 12/68

Calculate the sum

$$
\sum_{k=1}^{n} k \cdot\binom{n}{k}
$$
It is

$$
k \cdot\binom{n}{k}=k \frac{n!}{k!(n-k)!}=n \frac{(n-1)!}{(k-1)![(n-1)-(k-1)]!}=n \cdot\binom{n-1}{k-1}
$$

From this, it follows that

$$
\sum_{k=1}^{n} k\binom{n}{k}=n \cdot \sum_{k=1}^{n}\binom{n-1}{k-1}=n \cdot \sum_{l=0}^{n}\binom{n-l}{l} \cdot 1^{l} \cdot 1^{n-1-l}=n(1+1)^{n-1}=n \cdot 2^{n-1}
$$