Sum of Coeff: Evaluate 30C0 to 30C30

[chaoseverlasting]

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 don't know how to find the series or the coeff.


I don't know how to go about it.

 

[JasonRox]

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 don't know how to find the series or the coeff.


I don't 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]

Would those be the binomial coefficients? If so your ⁿC_i is more commonly written _nC_i.

 

[tim_lou]

is the thing alternating or what?

 

[drpizza]

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.

 

[chaoseverlasting]

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.

 

[tim_lou]

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

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

 

[chaoseverlasting]

Can someone work the first two steps or something and I can try to work the rest out?

 

[tim_lou]

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?