Two teams - Anadi and Khiladi- are playing a cricket match. Anadi requires 12 runs in the last 3 balls to win the match. Any team can score 0 to 6 runs, execpt 5 runs, from a particular ball. In how many different ways Team Anadi can score 12 runs without any no-ball and wide-ball in the last 3 balls?

Options

A. 11
B. 12
C. 13
D. none of these

Answer

To solve the problem of how many different ways Team Anadi can score 12 runs in the last 3 balls, we need to consider the possible scores for each ball. The possible scores for each ball are 0, 1, 2, 3, 4, and 6 runs (note that scoring 5 runs is not allowed).

We can represent the scores of the three balls as \( x_1, x_2, x_3 \) where \( x_1 + x_2 + x_3 = 12 \) and \( x_i \in \{0, 1, 2, 3, 4, 6\} \).

### Step 1: Identify possible combinations of scores
We need to find combinations of \( x_1, x_2, x_3 \) that add up to 12. We will consider different cases based on the maximum score of 6 runs on any ball.

### Case 1: Two balls score 6 runs
- \( x_1 = 6, x_2 = 6, x_3 = 0 \)
- Permutations: \( \frac{3!}{2!1!} = 3 \)

### Case 2: One ball scores 6 runs, and the other two score 4 and 2 runs
- \( x_1 = 6, x_2 = 4, x_3 = 2 \)
- Permutations: \( 3! = 6 \)

### Case 3: One ball scores 6 runs, and the other two score 3 runs each
- \( x_1 = 6, x_2 = 3, x_3 = 3 \)
- Permutations: \( \frac{3!}{1!2!} = 3 \)

### Case 4: All balls score 4 runs
- \( x_1 = 4, x_2 = 4, x_3 = 4 \)
- Permutations: \( \frac{3!}{3!} = 1 \)

### Step 2: Add all the cases
Now, we will add all the different ways we can achieve the score of 12 runs:
- From Case 1: 3 ways
- From Case 2: 6 ways
- From Case 3: 3 ways
- From Case 4: 1 way

Total ways = \( 3 + 6 + 3 + 1 = 13 \)

### Final Answer
Thus, the total number of different ways Team Anadi can score 12 runs in the last 3 balls is **13**.

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