Distributing marks to students - combinatorics problem

Question

30 marks are to be distributed to 8 students such that each student gets at least 2. In how many ways can this be done?

Answer 1

The problem is equal to distributing $14$ marks to $8$ students without any restriction, because as each student gets at least $2$, then $2 * 8 = 16$ of the marks should already be distributed equally.

Number of ways of distributing $n$ identical objects into $r$ groups is calculated by the formula:

$$C(n+r−1,r−1) = \frac{(n+r-1)!}{n! (r-1)!} $$

Then, by putting $n = 14 $ and $r = 8$, we have

$$C(14+8−1,8−1) = \frac{(14+8-1)!}{14! (8-1)!} $$

$$C(21,7) = \frac{(21)!}{14! 7!} $$

NOTE: I got help from the following question: Number of ways of distributing n identical objects among r groups

Answer 2

This is the stars and bars problem. Since each student gets $2$, we subtract out $2$ from each student to normalize. So each student gets at least no marks out of $14$ marks. The solution we get is $\binom{8 + 14 - 1}{14}$.