Sum of coeff.

chaoseverlasting

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

Evaluate:

[tex]^{30}C_0 ^{30}C_{10}-^{30}C1 ^{30}C_11+...+^{30}C_{20} ^{30}C_{30}[/tex]

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

Is this the whole question?

To evaluate it, wouldn't you actually need to know what the [itex]C_i[/itex] 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

[tex](-1)^n^{30}C_r ^{30}C_{10+r}[/tex] where r varies from 0 to 20.

tim_lou

hint:

1.[tex]\binom{a}{b}=\binom{a}{b-a}[/tex]

2. look at
[tex](1-x)^n(1+x)^n[/tex]

in general, when you have two series
[tex]S_1=\sum_n a_n x^n[/tex]
[tex]S_2=\sum_n b_n x^n[/tex]

[tex]S_1S_2=\sum_n\sum_{i+j=n} a_i b_jx^n[/tex]

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,

[tex]\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}[/tex]

look at the function
[tex]f(x)=\sum_{n}\left [\sum_{i=0}^{20} (-1)^n\binom{30}{i}\binom{30}{20-i}\right ]x^n[/tex]

from the equation I posted last time
[tex]S_1S_2=\sum_n\sum_{i+j=n}a_ib_jx^n=\sum_n\sum_{i=0}a_ib_{n-i}x^n[/tex]

what can you conclude?
what [itex]S_1[/itex] and [itex]S_2[/itex]
should you construct to finish the problem?