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# coefficient

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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