Question

coefficient of y3in expansion of
(2y+1)^4
​

Answer 1

Answer:

The multinomial theorem will tell you the following (I'll state it for trinomials)

(x1+x2+x3)n=∑k1+k2+k3=n(nk1,k2,k3)∏i=13xkii

In our case, we have x1=x, x2=−2y, x3=z. So if we are finding the coefficient of x2y2z4, note that this is equivalent to x21×x224×x43, which means we have to take k1=2,k2=2,k3=4. This gives us the answer (82,4,4)×4 (which comes because y2=x224, so we have to account for this):

8!×44!×2!×2!=1680

which should be correct (and is , I checked on Wolfram Alpha to make sure).

I case you are not aware of this theorem, read it up on Wikipedia. The proof is similar to that of the binomial theorem.

Step-by-step explanation: