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Find the coefficient of x^7 in the expan...

Find the coefficient of `x^7` in the expansion of `(1+3x-2x^3)^(10)dot`

Text Solution

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`(1+3x-2x^(3))^(10) = (10!)/(n_(1)!n_(2)!n_(3)!)(1)^(n_(1))(3x)^(n_(2)) - (-2x^(3))^(n_(3))`
where `n_(1) + n_(2) + n_(3) = 10` and `n_(2) + 3n_(3)) = 7`
The possible values of `(n_(1),n_(2),n_(3))` are `(3,7,0), (5,4,1),(7,1,2)`
`:.` Coefficient of `x^(7) = (10!)/(3! xx 7! xx 0!) xx (1)^(3) xx (3)^(7) xx (-2)^(0)`
` + (10!)/(3! xx 7! xx 0!) xx (1)^(3) xx (3)^(5) xx (-2)^(1)`
`+ (10!)/(7! xx 1! xx 2!) xx (1)^(7) xx (3)^(1) xx (-2)^(2)`
` = 26244 - 204120 + 4320`
` = - 173556`
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