## Sum of coeff.

1. The problem statement, all variables and given/known data

Evaluate:

$$^{30}C_0 ^{30}C_{10}-^{30}C1 ^{30}C_11+...+^{30}C_{20} ^{30}C_{30}$$

Again, I know this is some coeff of the product of some series, but I dont know how to find the series or the coeff.

I dont know how to go about it.
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 Quote by chaoseverlasting 1. The problem statement, all variables and given/known data Evaluate: $$^{30}C_0 ^{30}C_{10}-^{30}C1 ^{30}C_11+...+^{30}C_{20} ^{30}C_{30}$$ Again, I know this is some coeff of the product of some series, but I dont know how to find the series or the coeff. I dont know how to go about it.
Is this the whole question?

To evaluate it, wouldn't you actually need to know what the $C_i$ actually are?

Because if they're unknown constants, then I have no idea what is meant by evaluating that. Plus, your equation is not totally clear. You have a negative randomly put it where all the others are positives. You should explicitly write when the negatives are to be.
 Recognitions: Gold Member Science Advisor Staff Emeritus Would those be the binomial coefficients? If so your nCi is more commonly written nCi.

## Sum of coeff.

is the thing alternating or what?
 I'm assuming the thing is alternating.. Try working out the first few terms... Alternately, try to write a formula for the nth term, and a formula for the (n+1)th term. See what happens when you add them together.
 Yes, those are binomial coefficients. The general term comes out to be $$(-1)^n^{30}C_r ^{30}C_{10+r}$$ where r varies from 0 to 20.
 hint: 1.$$\binom{a}{b}=\binom{a}{b-a}$$ 2. look at $$(1-x)^n(1+x)^n$$ in general, when you have two series $$S_1=\sum_n a_n x^n$$ $$S_2=\sum_n b_n x^n$$ $$S_1S_2=\sum_n\sum_{i+j=n} a_i b_jx^n$$
 Can someone work the first two steps or something and I can try to work the rest out?
 reformulate the problem, basically you are asked to find, $$\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}$$ look at the function $$f(x)=\sum_{n}\left [\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}\right ]x^n$$ from the equation I posted last time $$S_1S_2=\sum_n\sum_{i+j=n}a_ib_jx^n=\sum_n\sum_{i=0}a_ib_{n-i}x^n$$ what can you conclude? what $S_1$ and $S_2$ should you construct to finish the problem?