Proving Binomial Identities: Sigma of k = 0 to m

In summary, the conversation discusses the expression (n, k) (n - k, m - k) = 2^m (n, m) and how to simplify it. The speaker demonstrates that (n, k) can be expanded to n!/k!(m-k)!(n-m)! and cancels out with (n-k, m-k), leaving n!/m!(n-m)!. They then realize that this is equal to (n, m) multiplied by the sum from k = 0 to m of 1, which simplifies to 2^m. The conversation concludes with the clarification that (n, k) (n - k, m - k) is not equal to 1/m!,
  • #1

i am supposed to show that

Sigma of k = 0 to m, (n, k) (n - k, m - k) = 2^m (n, m)

So I have after expanding:

(n, k) = n!/(n-k)!k! and (n-k, m-k) = (n-k)!/(m-k)!(n-m)!
so together the (n-k)! cancels out and I have
and that is
n!/m!(n-m)! which is
(n, m)

so then I can take (n, m) out of the sigma and then I would have
(n, m)*Sigma from k = 0 to m 1, but then how do I get the 2^m?

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  • #2
You seem to think that 1/(k!*(m-k)!) is 1/m!. It's not. Try some numerical examples. On the other hand it's easy to find the sum of that quantity over k. You probably proved sum over k of (m,k)=2^m, didn't you?
  • #3
Oh right. Thanks!

Yes, it's equal to (1 + 1) ^ m , so that is 2^m.

1. What is the purpose of proving binomial identities?

The purpose of proving binomial identities is to show that they are true for all values of the variables involved. This helps to establish the validity and reliability of these identities in various mathematical applications.

2. How do you approach proving a binomial identity?

To prove a binomial identity, you can use various techniques such as algebraic manipulation, mathematical induction, or the use of combinatorial arguments. It is important to carefully consider the given identity and choose the most appropriate method to prove it.

3. What is the role of sigma in binomial identities?

Sigma (Σ) is a Greek letter commonly used in mathematics to represent the summation of a series of terms. In the context of binomial identities, sigma is used to denote the sum of terms in the identity, which is typically represented by the variable k.

4. What is the significance of the limits in "sigma of k = 0 to m"?

The limits, k = 0 to m, indicate the range of values for the variable k that will be included in the summation. In the context of proving binomial identities, these limits help to specify the number of terms that need to be evaluated and provide a starting and ending point for the summation.

5. Can binomial identities be proven using other mathematical concepts?

Yes, binomial identities can be proven using a variety of mathematical concepts such as the binomial theorem, Pascal's triangle, and the binomial coefficients. These concepts provide alternative ways to express and manipulate binomial expressions, making it easier to prove their identities.