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

 Jan 1, 2023
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The coefficents of the expansion of a trinomial is a lot like what the binomial theorem states about the coefficients of a binomial. 

For example, if you want the coefficient of the a^3*b^2*c term of the expansion of (a+b+c)^6, it is the same as taking 3as, 2bs, and a c, and ordering them which is 6!/(3!*2!*1!). 

The thing is we don't know in this problem is how many y^2 or 2ys are multiplied together to create the y^4 term, so we can split this into 3 scenarios:

2 y^2 terms are multiplied together to get y^4

4 2y terms are multiplied together to get y^4

2 2ys and 1 y^2s are multiplied together to get y^4

The coefficent for the first case would be 6!/(2!*4!)*(-5)^4 = 15 * 625 = 9375

The coefficent for the second case would be 6!/(2!*4!)*2^4*(-5)^2 = 15*16*25 = 6000

The coefficent for the third case would be 6!/(2!*3!*1!)*2^2*(-5)^2 = 60*4*(-125) = -30000 

Adding all the possible cases together we get: -14625 is the coefficient of the y^4 term

 Jan 2, 2023

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