1. ## Counting Proof...

If n is a positive integer and n > 1, prove that C(n, 2) + C(n-1, 2) is a perfect square.

Any hints would be appreciated.

2. Hello, MasterShake!

I am not sure where to start with this problem. . . . . really?

If $\displaystyle n$ is a positive integer and $\displaystyle n > 1$,
prove that: .$\displaystyle C(n, 2) + C(n-1, 2)$ is a perfect square.

Do you know what those C-expressions mean?
If you do, just do the algebra . . .

$\displaystyle C(n,2) +C(n-1,2) \;=\;\frac{n!}{2!(n-2)!} + \frac{(n-1)!}{2!(n-3)!} \;=\;\frac{n(n-1)}{2} + \frac{(n-1)(n-2)}{2}$ . . . etc.

3. ## I was aware of nCr expresion

Even after trying to plug the equation I still am unsure how to set up the proof.

C(n, r) = n! / r!(n-r)!

shouldn't the bases of both equations be 2!(n-2)!

4. Originally Posted by MasterShake
Even after trying to plug the equation I still am unsure how to set up the proof.

C(n, r) = n! / r!(n-r)!

shouldn't the bases of both equations be 2!(n-2)!
No. In the second expression we have an n - 1 and r = 2, so (n - 1) - 2 = n - 3.

-Dan