A batsman can score 0, 2, 3, or 4 runs for each ball he receives. If N is the number of ways of scoring a total of 20 runs in one over of six balls, then N is divisible by :

Options

A. 5
B. 7
C. 14
D. 16

Answer

To solve the problem of finding the number of ways a batsman can score a total of 20 runs in one over (6 balls) with the possible scores of 0, 2, 3, or 4 runs per ball, we can use a combinatorial approach.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the combinations of scores (0, 2, 3, 4) that add up to 20 runs using exactly 6 balls.

2. **Setting Up the Equation**:
Let:
- \( x_0 \) = number of balls scoring 0 runs
- \( x_2 \) = number of balls scoring 2 runs
- \( x_3 \) = number of balls scoring 3 runs
- \( x_4 \) = number of balls scoring 4 runs

We have two equations:
- Total balls: \( x_0 + x_2 + x_3 + x_4 = 6 \)
- Total runs: \( 0 \cdot x_0 + 2 \cdot x_2 + 3 \cdot x_3 + 4 \cdot x_4 = 20 \)

3. **Rearranging the Equations**:
From the second equation, we can express it in terms of the number of balls:
\[ 2x_2 + 3x_3 + 4x_4 = 20 \]

4. **Finding Possible Combinations**:
We can iterate through possible values of \( x_4 \) (the number of balls scoring 4 runs) since it contributes the most to the total score:
- If \( x_4 = 5 \): \( 4 \cdot 5 = 20 \) (then \( x_2 = 0, x_3 = 0, x_0 = 1 \))
- If \( x_4 = 4 \): \( 4 \cdot 4 = 16 \) (then \( 2x_2 + 3x_3 = 4 \))
- If \( x_4 = 3 \): \( 4 \cdot 3 = 12 \) (then \( 2x_2 + 3x_3 = 8 \))
- If \( x_4 = 2 \): \( 4 \cdot 2 = 8 \) (then \( 2x_2 + 3x_3 = 12 \))
- If \( x_4 = 1 \): \( 4 \cdot 1 = 4 \) (then \( 2x_2 + 3x_3 = 16 \))
- If \( x_4 = 0 \): \( 0 \) (then \( 2x_2 + 3x_3 = 20 \))

5. **Calculating Each Case**:
For each case, we can find the number of combinations using the stars and bars method or multinomial coefficients.

6. **Summing Up All Combinations**:
After calculating the number of combinations for each case, we sum them up to find the total \( N \).

7. **Finding Divisibility**:
Finally, we check the divisibility of \( N \) by the given options.