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Coefficient of x34 in expansion of (1-3x+x2+2x4)10
By Slava Kisilevich on Monday, June 25, 2001 - 02:09 pm:

How do I find the coefficient of x34 of f(x), if f(x)=(1-3x+x2+2x4)10

Thanks

By The Editor:

When we multiply out 10 brackets, all of which are (1-3x+x2+2x4), we will get lots of terms, each of which is found my multiplying one term from each of the ten brackets.
So one x34 term will come from doing
2x4×2x4×2x4×2x 4×2x4×2x4×2x4×2x 4×x2×1.
Another will be
2x4×2x4×2x4×2x 4×2x4×2x4×2x4×x 2×2x4×1.
Another will be
2x4×2x4×2x4×2x 4×2x4×2x4×2x4×2x 4×(-3x)×(-3x).

Dan's reply shows you how to find all of the x34 terms.

By Dan Goodman on Saturday, June 30, 2001 - 02:41 am:

Work out all the ways of making 34 from adding up 10 of 0,1,2 and 4.
For example, 4+4+4+4+4+4+4+4+2+0.

Then work out how many different orderings of these symbols there are. In this case there are eight 4's, one 2 and one 0. So there are 10!/(8!)=90 different orderings.

For each of these orderings, we get 2x4.2x4.2x4.....2x4.x 2.1 = 28x34 = 256 x34.

So the coefficient of x34 gained from multiplying eight of the 2x4's an x2 and a 1 together is 90 × 256.

Now do this for the other ways of making 34 from 10 of 0,1,2 and 4 (another one is 4+4+4+4+4+4+4+4+1+1, and there is one more).

That wasn't too clearly explained, I hope you see what I'm getting at here.