A fair six-faced die is rolled 12 times. The | vedclass

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(C) The total number of outcomes when a die is rolled $12$ times is $6^{12}$.We want each of the $6$ faces to appear exactly $2$ times.This is a multi

Vedclass · Accepted Answer

$\frac{12!}{2^{6} 6^{12}}$

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(C) The total number of outcomes when a die is rolled $12$ times is $6^{12}$.We want each of the $6$ faces to appear exactly $2$ times.This is a multinomial distribution problem where we arrange $12$ items into $6$ groups of size $2$ each.The number of ways to distribute $12$ outcomes such that each face appears twice is given by the multinomial coefficient:$\frac{12!}{2! 2! 2! 2! 2! 2!} = \frac{12!}{(2!)^6} = \frac{12!}{2^6}$.Therefore,the required probability is:$P = \frac{12!}{2^6 	imes 6^{12}}$.

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$K$	$3K$	$5K$	$7K$	$8K$	$K$

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$0$	$k$	$2k$	$3k$	$3k^2$	$k^2$	$2k^2$	$7k^2+k$

$A$ fair six-faced die is rolled $12$ times. The probability that each face turns up exactly twice is equal to:

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