Question

If there are $n$ students and $r$ prizes $\left( r < n \right)$, then they can be given away:
(i) In ${{n}^{r}}$ ways when a student can receive any number of prizes.
(ii) In ${{n}^{r}}-n$ ways when a student cannot receive all the prizes.
A. True
B. False.

Answer 1

Hint: For solving this question you should know about the permutation and combinations. As we know that this problem has statements only and we have to tell whether these are true or false. So, first we will discuss the permutation and combination and then by applying the concept of these, we will find the solution of the problem and then we will find the existence of the given statements.

Complete step by step answer:
According to our question, if there are students and $r$ prizes $\left( rAs we know, permutations are ordered combinations. It means where the order does not matter, there it is a permutation. And permutations have also two types, one is repeated or repetition is allowed and the second is no repetition. The permutations with repetition are always the easiest to calculate. And the permutation with no repetition reduces the number of available choices each time. So, if we see our question, then:
Total ways to distribute the prize = ${{n}^{r}}$
This includes $n$ cases where anyone in particular receives all the prizes. So, ${{n}^{r}}-n$ ways when a student cannot receive all the prizes.

So, the correct answer is “Option A”.

Note: While solving this type of questions you have to ensure that we will use the permutation or combination here and, in the permutation, it is necessary to find out if there are any events that are repeating or not repeating.