1.

Find the number of integral solutions of x_(1)+x_(2)+x_(3)=24 subjected to the condition that 1 le x_(1) le 5, 12 le x_(2) le 18 " and" -1 le x_(3).

Answer»


Solution :Number of integral solutions
=coefficient of `p^(24) " in" (p+p^(2)+p^(3)+p^(4)+p^(5))`
`(p^(12)+p^(13)+..+p^(18))(p^(-1)+p^(0)+p^(1)+..+p^(11))`
=coefficient of `p^(24) " in" (p(1-p^(5)))/(1-p).(p^(12)(1-p^(7)))/(a-p).(p^(-1)(1-p^(13)))/(1-p)`
coefficient of `p^(12) " in" (1-p^(5))(1-p^(7))(1-p^(13))(1-p)^(-3)`
=coefficient of `p^(12) "in" (1-p^(5)-p^(7)+p^(12)(1-p)^(-3)`
`= .^(14)C_(12)- .^(9)C_(7)- .^(7)C_(5)+1`
=91-36-21+1=35


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