four fair dice `D_1, D_2,D_3 and D_4` each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously. The probability that `D_4` shows a number appearing on one of `D_1,D_2,D_3` is

Options

A. `(91)/(216)`
B. `(108)/(216)`
C. `(125)/(216)`
D. `(127)/(216)`

Answer

A

Explanation

As one of the dice shows a number appearing on one of `P_1, P_2 and P_3` .
Thus, three cases arise.
(i) All show same number.
(ii) Number appearing on `D_4` oppears on any one of `D_1,D_2 and D_3`
(iii) Number appearing on `D_4` appears on any two of `D_1,D_2 and D_3`
Sample space `=6xx6xx6xx6=6^4` favorite events =Case I or Case II or Case III
CaseI First we should select one number for `D_4` winch appears on all i.e `^6C_1xx1`
Case II For `D_4` there are `^6C_1` ways. Now, it appears on any one of `D_1,D_2 and D_3 i.e ""^3C_1xx1`
For other two there are `5xx5` ways
`rArr ^6C_1 xx^3C_1xx1xx5xx5`
For `D_4` there are `^6C_1` ways now it apperes on any two of `D_1,D_2 and D_3`
`rArr ^3C_2xx1^2`
For another one there are 5 ways
`rArr ""^6C_1xx""^3C_2xx1^2xx5`
Thus ,Probability `=(""^6C_1+""^6C_1xx^3C_1xx5^2+""^6C_1xx""^3C_2xx5)/(6^4)`
`=(6(1+75+15))/6^4`
`=91/216`