Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$ In the first course of probability by sheldon ross on page 155 at the bottom.
It is the proof for the expectation of Binomial distribution. It uses the identity:
$$i{n \choose i } =n {n-1 \choose i - 1}$$
Since learning combinatorics, I have been trying to think of this intuitively. So is there an intuitive proof for this identity, rather than just using brute force?
 A: Suppose you have a group of $n$ people, and you want to form a committee of $i$ people with one person as the chair of the committee. To count the number of ways to do this, you could first choose the $i$ people and then choose the chair from those $i$ people. This gives ${n \choose i} \cdot {i \choose 1}={n \choose i} \cdot i$ ways. Alternatively, you could first choose the chair of the committee and then choose the $i-1$ remaining committee members from the remaining $n-1$ people. This gives ${n \choose 1} \cdot {n-1 \choose i-1}=n \cdot {n-1 \choose i-1}$ ways.
A: Left hand side: Suppose you have $n$ people and want to choose $i$ people for a committee and from them one president. Then there are $n$ chose $i$ ways to choose the committee and $i$ ways to choose the president.
Right hand side: If we pick a president first, we have $n$ options. Then we need to pick $i-1$ more committee members from the remaining $n-1$ people.
Clearly both sides are the same thing, and you can see that you indeed get the expressions given.
