The number of ways in which the following prizes be given to a class of 20 boys, first and second mathematics, first and second physics , first chemistry and first english is

Options

A. `20^4 times19^2`
B. `20^3 times 19^3`
C. `20^2 times 19^4`
D. none of these

Answer

To solve the problem of distributing the prizes among 20 boys, we will break down the steps involved in calculating the total number of ways to award the prizes.

### Step-by-Step Solution:

1. **Identify the Prizes**:
- We have the following prizes to distribute:
- First and Second in Mathematics
- First and Second in Physics
- First in Chemistry
- First in English

2. **Calculate the Ways for Mathematics**:
- For Mathematics, we need to select 2 boys from 20 and assign them the first and second prizes. The order matters because one boy will receive the first prize and the other the second prize.
- The number of ways to choose and arrange 2 boys from 20 is given by the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \).
- Thus, the number of ways for Mathematics is:
\[ 20P2 = \frac{20!}{(20-2)!} = \frac{20!}{18!} = 20 \times 19 \]

3. **Calculate the Ways for Physics**:
- Similarly, for Physics, we also need to select 2 boys from 20 for the first and second prizes.
- The number of ways for Physics is:
\[ 20P2 = 20 \times 19 \]

4. **Calculate the Ways for Chemistry**:
- For Chemistry, we only need to select 1 boy for the first prize.
- The number of ways for Chemistry is:
\[ 20P1 = 20 \]

5. **Calculate the Ways for English**:
- For English, we also need to select 1 boy for the first prize.
- The number of ways for English is:
\[ 20P1 = 20 \]

6. **Combine All the Ways**:
- To find the total number of ways to distribute all the prizes, we multiply the number of ways for each subject:
\[ \text{Total Ways} = (20P2 \text{ for Math}) \times (20P2 \text{ for Physics}) \times (20P1 \text{ for Chemistry}) \times (20P1 \text{ for English}) \]
\[ = (20 \times 19) \times (20 \times 19) \times 20 \times 20 \]

7. **Simplify the Expression**:
- Now we can simplify:
\[ = (20^4) \times (19^2) \]

### Final Answer:
The total number of ways in which the prizes can be given is:
\[ 20^4 \times 19^2 \]