Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is _______.

GATE 2014 · Engineering Mathematics · Probability · medium

Answer: 10

1. Transform to shortfall variables: Let e_i = 6 - d_i for i = 1,2,3,4. Then e_i in {0,1,2,3,4,5} and e_1 + e_2 + e_3 + e_4 = 2.
2. Count non-negative integer solutions: n = 4 bins, r = 2: C(4 + 2 - 1, 2) = C(5, 2) = 10.
3. State result: X = 10. The probability that four dice sum to 22 is 10/1296.