First, let’s show the identity:
The prove procedure can be like this:
A problem in Probability theory is as follows: there are n kinds of color pens to paint a big piece of cloth with n rows ribbon, the goal is to assign colors to the ribbons, what’s the number of ways to paint k(k<=n) colors to the cloth?
let’s consider the situation when k=1, obvisouly,only one way to paint, i.e.,paint the cloth with the unique color. when k=2,we can obtain the result using the exclusive rule: using the whole possible ways dedute the numer of ways with only one color,thus 
similarly,we got
and with some simple algebra,we got 
as first introduced before.
and next ,we turned into the addictive expection rule in Probability theory, the question we put forwad is that how many colors are used in terms of average, more precisely what’s the expection of color used ?
before answering the question we denote X as a random variable as follows:
then the numer of color used can be derived as 
,and its expections is
and 
denoted by the probility of color i is used, from simple Probability theory,we know
thus,the right part of the identity is the expection of number of color used in the pianting (with replacement).up to now,what we have to prove is that the left part of the identity is also the expection, let’s dig it a little further, the left is a original calculation of the expection ,k colors used with possibility 
truly ,they are equal.
Asnebula's notes recorded 