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Question

An unbiased die, with faces numbered 1,2,3,4,5,6 is thrown n times and the list of n numbers showing up is noted. Then the probability that, among the numbers 1,2,3,4,5,6 only three numbers appear in this list ?

A
[3n3(2n)+3]6n
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B
6C3[3n3(2n)+3]6n
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C
6C3[3n3(2n)]6n
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D
None of these
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Solution

The correct option is B 6C3[3n3(2n)+3]6n
The total no. of outcomes = 6n
We can choose three numbers out of 6 in 6C3 ways. By using three numbers out of 6 we can get 3n sequences of length n. But these sequences of length n which use exactly two numbers and exactly one number.
The number of n – sequences which use exactly two numbers
= 3C2[2n1n1n]=3(2n2) and the number of n sequence which are exactly one number
= (3C1)(1n)=3
Thus, the number of sequences, which use exactly three numbers
= 6C3[3n3(2n2)3]=6C3[3n3(2n)+3]
∴ Probability of the required event,
= 6C3[3n3(2n)+3]6n

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