If there are n student and r prizes `(r lt n)` They can be given away
`n^r-n` ways when a student cannot receive all the prizes.True or False?

Answer

To determine whether the statement "If there are n students and r prizes (r < n), they can be given away in \( n^r - n \) ways when a student cannot receive all the prizes" is true or false, we can analyze the problem step by step.

### Step 1: Understanding the Problem
We have \( n \) students and \( r \) prizes, where \( r < n \). The statement suggests that we can distribute these prizes in \( n^r - n \) ways under the condition that no student can receive all the prizes.

### Step 2: Counting the Total Ways to Distribute Prizes
If there were no restrictions on how prizes could be distributed, each of the \( r \) prizes could be given to any of the \( n \) students. Therefore, the total number of ways to distribute \( r \) prizes among \( n \) students is:
\[ n^r \]
This is because for each prize, we have \( n \) choices (one for each student).

### Step 3: Accounting for the Restriction
The restriction states that no student can receive all \( r \) prizes. We need to subtract the cases where at least one student receives all the prizes.

### Step 4: Counting the Cases Where One Student Gets All Prizes
If one specific student receives all \( r \) prizes, there are \( n \) such students who could potentially take all the prizes. Therefore, there are \( n \) cases to subtract from our total.

### Step 5: Final Calculation
Thus, the number of valid distributions where no student receives all the prizes is:
\[ n^r - n \]

### Conclusion
Since we have derived that the number of ways to distribute the prizes under the given condition is indeed \( n^r - n \), we can conclude that the statement is **True**.