Summation of Products of Binomial Coefficients

In summary, the conversation discussed finding and proving a formula for the sum of combinations of r, s, and t with a fixed sum of n. The attempt at a solution involved trying to find the number of combinations that satisfy the given conditions, but the speaker suggested finding a combinatorial model to simplify the counting process.
  • #1
Vespero
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Homework Statement



Find and prove a formula for sum{ (m1 choose r)(m2 choose s)(m3 choose t) }

where the sum is over all nonnegative integers r, s, ant t with fixed sum r + s + t = n.

Homework Equations


The Attempt at a Solution



I first attempted to find the number of combinations of r, s, and t would satisfy r + s + t = n.
I found this to be (n+1)(n+2)/2. I have a feeling this is important (it gives the number of terms in the summation), but can't seem to find a way to apply it to find a formula.
 
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  • #2
Rather than trying to evaluate this sum using algebra, try to find a combinatorial model for it -- a scenario where the sum you have is obviously the number of different ways to do something. Then count that model in a simpler way.
 

FAQ: Summation of Products of Binomial Coefficients

1. What is the formula for calculating the summation of products of binomial coefficients?

The formula for calculating the summation of products of binomial coefficients is (n choose k)*(m choose k), where n and m are the number of terms in each binomial coefficient and k is the index of summation.

2. How is the summation of products of binomial coefficients related to the binomial theorem?

The summation of products of binomial coefficients is closely related to the binomial theorem, which states that for any real numbers x and y and a positive integer n, (x+y)^n = Σ (n choose k)*x^(n-k)*y^k. This means that the terms in the summation of products of binomial coefficients are coefficients in the binomial expansion of (x+y)^n.

3. What is the significance of the summation of products of binomial coefficients in mathematics?

The summation of products of binomial coefficients has many applications in mathematics, including in combinatorics, probability theory, and statistics. It is used to calculate the number of ways to choose a subset of objects from a larger set, as well as in the calculation of probabilities in various scenarios.

4. Can the summation of products of binomial coefficients be simplified or evaluated in a closed form?

Yes, in some cases the summation of products of binomial coefficients can be simplified or evaluated in a closed form. For example, when the binomial coefficients are symmetrical (n=m), the summation can be simplified to 2^n. However, in most cases, the summation cannot be simplified and must be evaluated numerically.

5. How is the summation of products of binomial coefficients used in real-world applications?

The summation of products of binomial coefficients has many real-world applications, including in the fields of computer science, economics, and engineering. It is used in the design and analysis of algorithms, in the calculation of expected values and probabilities in economic models, and in the estimation of error rates in communication systems.

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