A special dice with numbers 1,-1,2,-2,0 and 3 is thrown thrice. What is the probability that the sum of the numbers occurring on the upper face is zero?

Options

A. `1//72`
B. `1//8`
C. `7 // 72`
D. `25// 216`

Answer

D

Explanation

Total no.of elementar eveents ` = 6^(3)`.
Favourable no . Of elementry events.
= coefficient of `x^(0)` in `[x + x^(-1) + x^(0) + x^(-2) + x^(2) + x^(2) + x^(3)]^(3)`
= coeff .of `x^(0) "in" [(1+x+x^(2)+x^(3)+x^(4)+x^(5))/(x^(2))]`
=coeff.of `x^(6) "in"[(1-x^(6))/(1-x)]^(3)`
= coeff .of `x^(6)"in" [1-x^(6)]^(3)[1-x]`
= coeff.of `x^(6)` in `[ 1- x^(6)]^(3)[1-x]`
= coeff.of `x^(6)` in `[1-""^(3)C_(1)x^(6) }.....][1-x]^(-3)`
= coeff.of `x^(6)` in `(1-x)^(-3).""^(3)C_(1)` coeff.of `x^(0)` in `(1-x)^(-3)`
`=""^(6+3-1)C_(3-1) -""^(3)C_(1)`
` = ""^(8)C_(2) -""^(3)C_(1) = (8!)/(6!2!) - (3!)/(2!)`
`= (8 xx 7)/(2) -3= 25`.
Required probability `= (25)/(216)`.