Question

In how many ways 30 marks can be allotted to 8 questions if each question carries at least 2 marks

Answer 1

We start with $30$ marks. Since all questions must be worth at least $2$ marks, we take

$2\times 8=16$ marks from the $30$

and distribute them equally. Now each question has $2$ (so far) and the is left with $30-16=14$marks.

Now we just need to find the number of ways to divide up the remaining $14$ marks among the $8$ questions, we calculate

${14+7}_{C_{14}}={21}_{C_{14}}=\frac{21!}{14!7!}$

So there are $116,280$ ways to assign $30$ marks to $8$ questions, where each question is worth at least $2$ marks.