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Question
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
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Solution
The given expression is `(x^4 - 1/x^3)^15`
General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^15"C"_r (x^4)^(15 - r) (- 1/x^3)^r`
= `""^15"C"_r (x)^(60 - 4r) (-1)^r * 1/x^(3r)`
= `""^15"C"_r (-1)^r * 1/(x^(3r - 60 + 4r))`
= `""^15"C"_r (-1)^r * 1/(x^(7r - 60))`
To find the coefficient of `1/x^17`
Put 7r – 60 = 17
⇒ 7r = 60 + 17
⇒ 7r = 77
∴ r = 11
Putting the value of r in the above expression, we get
= `""^15"C"_11 (-1)^11 * 1/x^17`
= `- ""^15"C"_4 * 1/x^17`
= `- (15 xx 14 xx 13 xx 12)/(4 xx 3 xx 2 xx 1) * 1/x^17`
= `- 1365 * 1/x^17`
Hence, the coefficient of `1/x^17` = – 1365
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