In how many ways can 10 different prizes be given to five students if the head boy must get 4 prizes and rest of the students can get any number of prizes?

Options

A. a. 270
B. b. 360
C. c. 720
D. d. none of these

Answer

To solve the problem of distributing 10 different prizes to 5 students with the condition that the head boy must receive 4 prizes, we can follow these steps:

### Step 1: Select Prizes for the Head Boy
We need to choose 4 prizes out of the 10 available prizes for the head boy. The number of ways to choose 4 prizes from 10 is given by the combination formula:

\[ \text{Number of ways to choose 4 prizes} = \binom{10}{4} \]

Calculating this, we have:

\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

### Step 2: Distribute Remaining Prizes
After giving 4 prizes to the head boy, we have 6 prizes left (10 - 4 = 6). These 6 prizes can be distributed among 5 students (including the head boy) in any manner.

Each of the 6 remaining prizes can go to any of the 5 students. Therefore, for each prize, there are 5 choices.

The total number of ways to distribute the 6 prizes is:

\[ \text{Number of ways to distribute 6 prizes} = 5^6 \]

Calculating this, we have:

\[ 5^6 = 15625 \]

### Step 3: Total Ways to Distribute Prizes
Now, to find the total number of ways to distribute the prizes, we multiply the number of ways to choose the prizes for the head boy by the number of ways to distribute the remaining prizes:

\[ \text{Total ways} = \binom{10}{4} \times 5^6 = 210 \times 15625 \]

Calculating this gives:

\[ 210 \times 15625 = 3281250 \]

### Final Answer
Thus, the total number of ways to distribute the 10 different prizes to the 5 students, ensuring the head boy receives 4 prizes, is:

\[ \boxed{3281250} \]