Binomial Theorem coefficients proof

In summary: RlIHRoaXMgcGFydCBiZSBhIG1vcmUgZWxhZ2Ugd2l0aCBkZWZpbmVkIChtIGsuKSBpcyBzaG93aW5nIHRoYXQgKG4ga3MpIHJlcXVpcmUgaWYgKGRlZmluZWQpIGlzIHZlcnkgb25lIGNvbnRleHRzLiBUaGF0IGlzIHNpemUgb2YgdGhlIHJlcXVpcmUgaXNzdWVzIHRoYXQgdGhlIHByb2
  • #1
srfriggen
306
5

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.

 
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  • #2
srfriggen said:

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.
 
  • #3
srfriggen said:
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
 

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that provides a way to expand a binomial expression raised to a positive integer power. It is used to find the coefficients of each term in the expansion.

2. What are the coefficients in the Binomial Theorem?

The coefficients in the Binomial Theorem are the numbers that appear in front of each term in the expansion. They represent the number of ways that the terms can be combined to form the expression.

3. How do you prove the coefficients in the Binomial Theorem?

The coefficients in the Binomial Theorem can be proven using mathematical induction. The proof involves showing that the formula for the coefficients holds true for the base case (n = 1) and then using the assumption that it holds true for n = k to prove that it also holds true for n = k+1.

4. Why is the Binomial Theorem important?

The Binomial Theorem has many applications in mathematics, statistics, and engineering. It is used to solve problems involving binomial distributions, probability, and series expansions. It also provides a way to simplify complex expressions and make calculations easier.

5. What if the exponent in the Binomial Theorem is a negative number?

The Binomial Theorem can still be applied if the exponent is a negative number. In this case, the expression is expanded using negative binomial coefficients. These coefficients are calculated using the formula: (-1)^k * (n+k-1 choose k).

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