TITLE: Evaluate: $ \sum _ {k=1}^{n}{\frac{k}{n}\binom{n}{k}t^k(1-t)^{n-k}}$
QUESTION [5 upvotes]: Evaluate:
$$ \sum _ {k=1}^{n}{\frac{k}{n}\dbinom{n}{k}t^k(1-t)^{n-k}} $$
$\dbinom{n}{k}$stands for the usual binomial coefficient giving the number of
ways of choosing $k$ objects from n objects.

Totally stuck.How can I able to solve this?

REPLY [13 votes]: Using
$$
    \binom{n}{k} \frac{k}{n} = \binom{n-1}{k-1}
$$
Then
$$
   \sum_{k=1}^n \frac{k}{n} \binom{n}{k} t^k (1-t)^{n-1} = \sum_{k=1}^{n} \binom{n-1}{k-1} t^{k} (1-t)^{n-k} = t \sum_{k=0}^{n-1} \binom{n-1}{k} t^{k} (1-t)^{n-1-k} = (t+1-t)^{n-1} t = t
$$