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Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $\dfrac{X}{1296}$. The value of $X$ is _______

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Using IEP method.

Why we r not thinking this way -

(6,6,6,4) OR (6,6,5,5)

So probability will be:

(1/6 * 1/6 * 1/6 * 1/6) + (1/6 * 1/6 * 1/6 * 1/6) = 2/1296

So, value of X is 2
by

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6 6 6 4 can occur in 4C3 = 4 ways. (6,6,6,4), (6,6,4,6), (6,4,6,6), and (4,6,6,6)

Similarly, 6,6,5,5 can occur in 4C2 = 6 ways

So, 6+4 = 10.