Problem with Combinatorics: C_{k+1}^m + C_{k-1}^m + 2 C_k^m = C_{k+1}^{m+2}

[dunkmaster]

Hi everyone. This is my first post and I hope you can help me solve this problem I'm having. Anyway this it the problem. I've been doing combinatorics for about a week at school and I'm having an exam soon(I'm not asking you to help me understand it, not solve it for me). Problem is, "hardest" thing that will be on test is this type :

. . . . .\(\displaystyle \large{ C_{k+1}^m\, +\, C_{k-1}^m\, +\, 2\, C_k^m\, =\, C_{k+1}^{m+2} }\)

Stuff I've been working with is kinda like this:

. . . . .\(\displaystyle \large{ V_n^3\, +\, C_n^{n-2}\, =\, 14n }\)

I just dont know how to solve this type of problem. I can't find a way to solve it with m and k in the game. Problem is that my teacher said that we need to find that out by ourselves and that can cost me the grade. Please help me solve this.

 

[Deleted member 4993]

Hi everyone. This is my first post and I hope you can help me solve this problem I'm having. Anyway this it the problem. I've been doing combinatorics for about a week at school and I'm having an exam soon(I'm not asking you to help me understand it, not solve it for me). Problem is, "hardest" thing that will be on test is this type :

. . . . .\(\displaystyle \large{ C_{k+1}^m\, +\, C_{k-1}^m\, +\, 2\, C_k^m\, =\, C_{k+1}^{m+2} }\)

Stuff I've been working with is kinda like this:

. . . . .\(\displaystyle \large{ V_n^3\, +\, C_n^{n-2}\, =\, 14n }\)

I just dont know how to solve this type of problem. I can't find a way to solve it with m and k in the game. Problem is that my teacher said that we need to find that out by ourselves and that can cost me the grade. Please help me solve this.

\(\displaystyle \displaystyle{C^m_{k+1} \ = \ \dfrac{m!}{(m-k-1)! * (k+1)!}}\)

In the similar fashion:

\(\displaystyle \displaystyle{C^m_{k-1} \ = \ ??}\)

and

\(\displaystyle \displaystyle{2 * C^m_{k} \ = \ ??}\)

and continue.....