Binomial Theorem coefficients proof

[srfriggen]

Homework Statement

Define (n k) = n!/k!(n-k)! for k=0,1,...,n.

Part (b) Show that (n k) + (n k-1) = (n+1 k) for k=1,2,...n.

Part (c) Prove the binomial theorem using mathematical induction and part (b).

Homework Equations

The Attempt at a Solution

I'm wasn't able to find the correct symbols to write out what (n k) should look like (it should be vertical). I hope it is still clear what was meant. If anyone knows what symbols to use on this site please let me know.


I attempted this by using induction, but it got pretty sloppy pretty quickly. Before I try to pursue that route I was wondering if there was a more elegant way to approach the problem. Or if using induction, what would be the best way to start. Perhaps if someone can lay out the approach I should be able to make a good dent in the problem.

I will post any questions about part (c) only after I solve part (b).

Thank you for reading.

 

[srfriggen]

Homework Statement

Define (n k) = n!/k!(n-k)! for k=0,1,...,n.

Part (b) Show that (n k) + (n k-1) = (n+1 k) for k=1,2,...n.

Part (c) Prove the binomial theorem using mathematical induction and part (b).

Homework Equations

The Attempt at a Solution

I'm wasn't able to find the correct symbols to write out what (n k) should look like (it should be vertical). I hope it is still clear what was meant. If anyone knows what symbols to use on this site please let me know.


I attempted this by using induction, but it got pretty sloppy pretty quickly. Before I try to pursue that route I was wondering if there was a more elegant way to approach the problem. Or if using induction, what would be the best way to start. Perhaps if someone can lay out the approach I should be able to make a good dent in the problem.

I will post any questions about part (c) only after I solve part (b).

Thank you for reading.

I just realized how to do this part (b). Please disregard.

 

[Ray Vickson]

I just realized how to do this part (b). Please disregard.

OK, but you also asked about notation. Some standard notations that do not use LaTeX are: C(n,m) or _nC_m; you get this last one by using the "X₂" button in the menu at the top of the input panel. If you want the full, fancy version using LaTeX, you can say "[it e x] {n \choose m} [/i t e x]" for an in-line version or
"[t e x] {n \choose m} [/t e x]" for a displayed version. Remove all spaces in [] and remove the quotation marks. These give ${n \choose m}$ and
$${n \choose m},$$ respectively.

RGV