Question

Six dice are thrown simultaneously. The probability that exactly three of them show the same face and ramining three show different faces, is
A. `((5!)^(2))/(6^(5))`
B. `(5!)/(2!6^(6))`
C. `((5!)^(2))/(2(6^(6))`
D. `(5!)/(2(6^(6)))`

Answer 1

Correct Answer - C
Six dice when thrown simultaneously can result in `6^(6)` ways.
`therefore` Total number of elementary events `=6^(6)`.
Select a number which occurs on three dice out of six numbers 1,2,3,4,5,6. This can be done in `.^(6)C_(1)` ways. Now, select three numbers out of the remaining 5 numbers. This can be done in `.^(5)C_(3)` ways. Now, we have 6 numbers like 1,2,3,4,4,4, 2,3,6,1,1,1 etc. These digits can be arranged in `(6!)/(3!)` ways.
So, the number of ways in which three dice show the same face and the remaining three show distinct faces is
`.^(6)C_(1)xx .^(5)C_(3)xx(6!)/(3!)`
`therefore` Favourable number of elementary events `= (.^(6)C_(1)xx .^(5)C_(3)xx(6!)/(3!))/(6^(6))=((5!)^(2))/(2(6^(6)))`