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 :
i) 5
ii) 7
iii) 14
iv) 16

Options

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

Answer

To solve the problem, we need to find 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 for each ball. We will analyze different combinations of these scores that add up to 20 runs.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The batsman can score 0, 2, 3, or 4 runs on each ball. We need to find combinations of these scores that total 20 runs over 6 balls.

2. **Case 1: Five 4s and One 0**:
- If the batsman scores 4 runs on 5 balls and 0 on 1 ball, the combination would be:
- Runs: 4, 4, 4, 4, 4, 0
- The number of ways to arrange these scores is given by the formula for permutations of multiset:
\[ \text{Ways} = \frac{6!}{5! \cdot 1!} = 6 \]

3. **Case 2: Four 4s and Two 2s**:
- If the batsman scores 4 runs on 4 balls and 2 runs on 2 balls, the combination would be:
- Runs: 4, 4, 4, 4, 2, 2
- The number of ways to arrange these scores is:
\[ \text{Ways} = \frac{6!}{4! \cdot 2!} = 15 \]

4. **Case 3: Two 4s and Four 3s**:
- If the batsman scores 4 runs on 2 balls and 3 runs on 4 balls, the combination would be:
- Runs: 4, 4, 3, 3, 3, 3
- The number of ways to arrange these scores is:
\[ \text{Ways} = \frac{6!}{2! \cdot 4!} = 15 \]

5. **Case 4: Three 4s, Two 3s, and One 2**:
- If the batsman scores 4 runs on 3 balls, 3 runs on 2 balls, and 2 runs on 1 ball, the combination would be:
- Runs: 4, 4, 4, 3, 3, 2
- The number of ways to arrange these scores is:
\[ \text{Ways} = \frac{6!}{3! \cdot 2! \cdot 1!} = 60 \]

6. **Total Number of Ways (N)**:
- Now, we sum all the ways from each case:
\[ N = 6 + 15 + 15 + 60 = 96 \]

7. **Divisibility Check**:
- Now we need to check the divisibility of 96 by the given options:
- **i)** 5: 96 is not divisible by 5.
- **ii)** 7: 96 is not divisible by 7.
- **iii)** 14: 96 is not divisible by 14.
- **iv)** 16: 96 is divisible by 16.

Thus, the answer is that \( N \) is divisible by **16**.