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Find the coefficient of y^4 in the expansion of (2y-5+3y^2)^6.

 Jul 20, 2023
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There are 3 ways to get a \(y^4\) term: \((2y)^4\)\((3y^2) ^2\), and \((2y)^2 \times (3y^2)\)

 

For the first way, we need 4 \(2y\)'s  and 2 \(-5\)'s, so we have\((2y)^4 \times (-5)^2 = 400y^4\). But we also need to multiply by \({6 \choose 2} = 15\) because there are 15 ways to order them. This makes for \(400y^4 \times 15 = 6000y^4\)

 

For the second way, we need 2  \(3y^2\)'s and 4 \(-5\)'s, so we have \((3y^2)^2 \times (-5)^4 = 5625y^4\). But like last time, we need to multiply by 15 for the same reason. Thus we have \(5625y^4 \times 15 = 84375y^4\)

 

For the final way, we need 2 \(2y\)'s, 1 \(3y^2\)'s, and 3 \(-5\)'s, so we have \((2y)^2 \times (3y^2) \times (-5)^3 = -1500y^3\). But this time, we need to multiply by \({6! \over 3!2!} = 60\). Thus we have \(-1500y^4 \times 60 = -90000y^4\)

 

So in total, the coefficient of \(y^4\) is \(6,000y^4 + 84,375y^4 - 90,000y^4 = \color{brown}\boxed{375y^4}\)

 Jul 20, 2023

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