Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
You are not logged in.
Post a reply
Topic review (newest first)
- cooljackiec
- 2013-01-02 07:09:54
Algebraically?
- anonimnystefy
- 2013-01-02 06:17:24
Another way:
Write the sum on the left as:
We can notice that the number on the RHS is the number of ways to choose r+1 numbers out of the set
The sum on the LHS can be interpreted like this:
If we know that the greatest number we will choose is r+k (for ), then we can choose the rest of the numbers in ways. If we sum all those values for all different values of k between 1 and n-r+1, we will get the total number of ways to choose r+1 numbers from the set mentioned before.
But, we also know that that will be the sum on the RHS, so the RHS and the LHS must be equal!
- scientia
- 2013-01-02 05:40:09
You can check that
(Expand LHS and show it's equal to RHS.) Hence
Adding up …
Rearrange, noting that , and you're done.
- cooljackiec
- 2013-01-02 05:06:34
how would we prove the identity:
in algebra?