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Find the coefficient of $y^4$ in the expansion of $(2y-5)^6$.

 May 12, 2023
 #1
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By the Binomial Theorem,

[(2y-5)^6 = \binom{6}{0}(2y)^6(-5)^0 + \binom{6}{1}(2y)^5(-5)^1 + \binom{6}{2}(2y)^4(-5)^2 + \binom{6}{3}(2y)^3(-5)^3 + \binom{6}{4}(2y)^2(-5)^4 + \binom{6}{5}(2y)^1(-5)^5 + \binom{6}{6}(2y)^0(-5)^6.]

The coefficient of y4 is \(\binom{6}{4} (2y)^2 (-5)^2 = \boxed{480}\).

 May 12, 2023
 #2
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Expand:    (2y  -  5)^6

 

64 y^6 - 960 y^5 + 6000 y^4 - 20000 y^3 + 37500 y^2 - 37500 y + 15625

 

Coefficient of y^4 == + 6,000

 May 13, 2023

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