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.

Answer

B, C

Explanation

Let us define a onto function F from A : `[ r_1 r_2....r_n]` to B : [1,2,3] where , `r_1 ,r_2....r_n` are the readings of n throws and 1, 2, 3 are the numbers that appear in the n throws.
Number of such functions, M=N -[ n(1) -n(2)+n(3)]
where, N = total number of functions and n(t) = number of function having exactly clements in the range.
Now `N=3^n ,n(1)=3 .2^n, n(2) =3, n (3)=0 `
`rArr M= 3^n - 3.2^n + 3 `
Hence, the total'number of favourable cases
`=(3^n - 3.2 ^n +3 ). ""^(6)C_3`
`therefore ` Reqmred probability `=((3^n-3.2^n+3)xx""^(6)C_3)/(6^n)`