# GATE | GATE-CS-2014-(Set-1) | Question 65

Four fair six-sided dice are rolled. The probability that the sum being 22 is X/1296. The value of X is ________
(A) 7
(B) 8
(C) 9
(D) 10

Explanation: In general, Probability (of an event ) = No of favorable outcomes to the event / Total number of possible outcomes in the random experiment.

Here, 4 six-faces dices are tossed, for one dice there can be 6 equally likely and mutually exclusive outcomes.

Taking 4 together, there can be total number of 6*6*6*6 = 1296 possible outcomes.
Now, No of favorable cases to the event : here event is getting sum as 22.

So, there can be only 2 cases possible.

Case 1: Three 6’s and one 4, for example: 6,6,6,4 ( sum is 22)
Hence, No of ways we can obtain this = 4!/3! = 4 ways ( 3! is for removing those cases where all three 6 are swapping among themselves)

Case 2: Two 6’s and two 5’s,for example: 6,6,5,5 ( sum is 22)
Hence, No of ways we can obtain this = 4! /( 2! * 2!) = 6 ways ( 2! is for removing those cases where both 6 are swapping between themselves, similarly for both 5 also)

Hence total no of favorable cases = 4 + 6 = 10.
Hence probability = 10/1296.
Therefore option D.

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