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In how many ways te sum of upper faces of four distinct dices can besix. |
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Answer» 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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