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In how many ways the sum of upper faces ...

In how many ways the sum of upper faces of four distinct dices can be six.

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Let the numbers on the upper faces of four distinct dice be `x_(1),x_(2),x_(3) " and " x_(4)`.
The sum of the numbers is 6.
`therefore x_(1)+x_(2)+x_(3)+x_(4)=6, " " "where" 1 le x_(1),x_(2),x_(3),x_(4) le 6`
`therefore ` Number of ways the sum of the upper faces of dice is six
=coefficient of `p^(6) " in" (p+p^(2)+p^(3)+p^(4)+p^(5)+p^(6))^(4)`
=coefficient of `p^(6) " in" p^(4)(1+p+p^(2)+p^(3)+p^(4)+p^(5))^(4)`
coefficient of `p^(2) " in" (1+p+p^(2)+p^(3)+p^(4)+p^(5))^(4)`
=coefficient of `p^(2) " in" (1+p+p^(2)+.. " infinite terms")^(4)` (as terms with higher powers of p are not conisdered while calculating coefficient of `p^(2))`
= coefficient of `p^(2) " in" ((1)/(1-p))^(4)`
= coefficient of `p^(2) " in " (1-p)^(4)`
`= .^(4+2-1)C+_(2)`
`= .^(5)C_(2)=10`
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