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# I need help

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I know there's a problem like this already posted, but the answer was wrong

Find the positive integer n such that the expansion of \((4x^2 - 7y^3)^n\) contains a term of the form \(cxy^6\)(the x in this is actually to the power of 10 as in x^10, but the LaTeX didn't work).

Jun 17, 2023

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Don't you have to know the exact term such as: x^4 y^6, because x and y appear together in every term except in the first term and in the last term and c would be the coefficient.

Jun 17, 2023
#2
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We expand (4x2−7y3)n using the Binomial Theorem. The term of the form cx10y6 will come from the term in the expansion that has 10 factors of x and 6 factors of y. The coefficient of this term is C(n,10)*4^10*-7^6

For this term to be non-zero, we must have n≥10. Also, for this term to be the largest power of x in the expansion, we must have 10 of the n factors of x come from the 4x^2 term, and the other 2 factors of x must come from the other 2x^2 terms in the expansion. This means that we must have n≥10 and n≡2(mod3).

The smallest positive integer that satisfies both of these conditions is n=11​.

Jun 17, 2023