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Binomial Theorem coefficients proof

  • Thread starter srfriggen
  • Start date
  • #1
288
3

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.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
288
3

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.

Homework Statement





Homework Equations





The Attempt at a Solution

I just realized how to do this part (b). Please disregard.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
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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 nCm; 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 [itex] {n \choose m}[/itex] and
[tex] {n \choose m},[/tex] respectively.

RGV
 

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