Three six faced fair dice are thrown together.The probability that the sum of the numbers appearing on the dice is `k(3lekle8),` is

Options

A. `(k^(2))/(432)`
B. `(k(k-1))/(432)`
C. `((k-1)(k-2))/(432)`
D. `(k(k-1)(k-2))/(432)`

Answer

C

Explanation

The total number of elementary events associated to the random experiment of throwing three dice is `6xx6xx6=6^(3)`.
Favourable number of elementary events
=Coeff. Of `x^(k) " in " (x+x^(2)+x^(3)+..+x^(6))^(3)`
=Coeff. Of `x^(k-3) " in " ((1-x^(6))/(1-x))^(3)`
= Coeff. of `x^(k-3) " in " (1-x^(6))^(3)(1-x^(-3)`
= Coeff. of `x^(k-3) " in " (1-x)^(-3) " " [because 0 le k-3 le 5]`
`=.^(k-3+3-1)C_(3-1) [ because " Coeff.of" x^(n) " in " (1-x)^(-r) = .^(n+r-1)C_(r-1)]`
`= .^(k-1)C_(2) =((k-1)(k-2))/(2)`.
`therefore` Requried probability `=((k-1)(k-2))/(2xx6^(3))=((k-1)(k-2))/(432)`