Sum of coeff.

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.

 

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

 

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.

 

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]

 

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?