# Sum of coeff.

## Homework Statement

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.

JasonRox
Homework Helper
Gold Member

## Homework Statement

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.

HallsofIvy
Homework Helper
Would those be the binomial coefficients? If so your nCi is more commonly written nCi.

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
$$S_1=\sum_n a_n x^n$$
$$S_2=\sum_n b_n x^n$$

$$S_1S_2=\sum_n\sum_{i+j=n} a_i b_jx^n$$

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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?

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