Combination with repetitions. The formula for computing a k-combination with repetitions from n elements is:
$$\binom{n + k - 1}{k} = \binom{n + k - 1}{n - 1}$$
I would like if someone can give me a simple basic proof that a beginner can understand easily.

To make the things more clear let there are $n$ different kinds of drinks available in a restaurant, and there are $m$ people including yourself, at your birthday party. Each one is to be served only one 'drink' as one desires/orders without restriction that two or more can ask for different or the same drink as per individual wish.
Obviously, there can be at the most $m-1$ people asking for a brand of the 'drink' which have already been desired by other people. Thus at most $m-1$ copies of one or more brands. Let $i$ people desired different brands out of the available $n$ brands which they can ask for in $n\choose i$ ways, and $i$ can vary from $1$ to $m$. Further more, the remaining $m-i$ people must have desired the same brand(s) as already opted by the $i$ people. This can be done in $m-1\choose m-i$ combinations. Thus, the drinks that can be served, have ${n\choose i}{m-1\choose m-i}$ options. Hence total number of options is the summation of ${n\choose i}{m-1\choose m-i}$ over $i$. After using the Vandermonde's Identity.
This is equal to $n+m-1\choose m$.
Therefore, the total number of options is ${n+m-1\choose m}= \frac{(n+m-1)!}{m!(n-1)!}$.
So, if $n=5$ and $m=3$ then total number of options is given by
$= \frac{(5+3-1)!}{3!(5-1)!} = \frac{7!}{3!4!} = 35$
