Question

If 5 dice are rolled simultaneously, then the number of ways of getting a total of 7 from the numbers on their faces is:

• A. \( 12 \)
• B. \( 15 \)
• C. \( 20 \)
• D. \( 25 \)

Hint:

To count dice sums: transform to a "stars and bars" partitioning problem by adjusting for minimum face value.

Answer 1

The Correct Option is B

Step 1: Use integer partition method for dice We are looking for the number of non-negative integer solutions to: \[ x_1 + x_2 + x_3 + x_4 + x_5 = 7 \] where each \( x_i \geq 1 \) and \( \leq 6 \). Convert to: \[ y_1 + y_2 + y_3 + y_4 + y_5 = 2 \text{ (where } y_i = x_i - 1 \geq 0\text{)} \] Now number of integer solutions of \( y_1 + y_2 + y_3 + y_4 + y_5 = 2 \) is: \[ \binom{2 + 5 - 1}{4} = \binom{6}{4} = 15 \]