# Binomial Series

## The Attempt at a Solution

Is there any difference between the above expression and
?

Is there any relation between these two?

VietDao29
Homework Helper

## Homework Statement

Are you sure there are up to 2 sigma signs in that expression? By the way, you mean $$C_r^n$$ right?

If there's just one sigma, then $$\sum_{0 \le r < s \le n} (C_r^n + C_s^n)$$ is different from $$\sum_{r = 0}^n \sum_{s = 0}^n (C_r^n + C_s^n)$$.

In the first sum $$\sum_{0 \le r < s \le n} (C_r^n + C_s^n)$$, r, and s can take any value raging from 0 to n, but r must be less than s.

However, in the second sum: $$\sum_{r = 0}^n \sum_{s = 0}^n (C_r^n + C_s^n)$$, r, and s can take any value raging from 0 to n, no more requirement is needed.

So, in general, the second sum will have more terms than the first sum.

tiny-tim
Homework Helper
Hi Abdul!

The second one is roughly double the first, since it contains eg C1 + C2 but not C2 + C1.

hmm … what about all the terms such as C1 + C1 ?

can you find an exact equation for the difference between the second and twice the first?

Are you sure there are up to 2 sigma signs in that expression?

Yeah there are 2 sigma signs. 0<=r<s<=n is in between the two sigma signs.

can you find an exact equation for the difference between the second and twice the first?

Does that equate to
?

tiny-tim