A batsman can score `0, 1, 2, 3, 4` or `6` runs from a ball. The number of different sequences in which he can score exactly 30 runs in an over of six balls

Options

A. 4
B. 72
C. 56
D. 71

Answer

To solve the problem of finding the number of different sequences in which a batsman can score exactly 30 runs in an over of six balls, we will follow these steps:

### Step 1: Identify Possible Scores
The batsman can score the following runs from a ball: 0, 1, 2, 3, 4, or 6. We need to find combinations of these scores that add up to exactly 30 runs over 6 balls.

### Step 2: List Combinations
We will list the possible combinations of scores that total 30 runs:

1. **Combination 1**: 6, 6, 6, 6, 6, 0 (five 6s and one 0)
2. **Combination 2**: 6, 6, 6, 6, 4, 2 (four 6s, one 4, and one 2)
3. **Combination 3**: 6, 6, 6, 4, 4, 4 (three 6s and three 4s)
4. **Combination 4**: 6, 6, 4, 4, 3, 3 (two 6s, two 4s, and two 3s)

### Step 3: Calculate the Number of Ways for Each Combination
Now we will calculate the number of different sequences for each combination using the formula for permutations of a multiset:

1. **For Combination 1**:
- Sequence: 6, 6, 6, 6, 6, 0
- Number of ways = \( \frac{6!}{5! \cdot 1!} = 6 \)

2. **For Combination 2**:
- Sequence: 6, 6, 6, 6, 4, 2
- Number of ways = \( \frac{6!}{4! \cdot 1! \cdot 1!} = 30 \)

3. **For Combination 3**:
- Sequence: 6, 6, 6, 4, 4, 4
- Number of ways = \( \frac{6!}{3! \cdot 3!} = 20 \)

4. **For Combination 4**:
- Sequence: 6, 6, 4, 4, 3, 3
- Number of ways = \( \frac{6!}{2! \cdot 2! \cdot 2!} = 90 \)

### Step 4: Sum the Ways
Now, we will sum the number of ways from all combinations:

- Total ways = \( 6 + 30 + 20 + 90 = 146 \)

### Final Answer
The total number of different sequences in which the batsman can score exactly 30 runs in an over of six balls is **146**.