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An unbiased dice, with faces numbered 1,...

An unbiased dice, with faces numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n numbers shown up is noted. Then find the probability that among the numbers 1, 2, 3, 4, 5, 6 only three numbers appear in this list and each number appears at least once.

Text Solution

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When a dice is rolled n times, total number of cases is `6^(n)` Let us first select three numbers which appear.
Three numbers can be selected from six `.^(6)C_(3)` ways. Now each of these three numbers appear at least once. This is same as filling three different boxes with n different objects such that no box remains empty.
Number of ways of such distribution using the principle of inclusion-exclusion = `3^(n) - .^(3)C_(1)(3 - 1)^(n ) + .^(3)C_(2) (3 - 2)^(n)`
`= 3^(n) - 3 xx 2^(n) + 3`
`therefore` Required probability = `((3^(n) - 3 xx 2^(n) + 3) xx .^(6)C_(3))/(6^(n))`
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