Sum of combinations from k to n

[Ediliter]

I have been trying to figure out a formula for the sum of combinations. For example:

[itex]\sum[/itex]ⁿ_k=0([itex]\frac{n}{k}[/itex]) = 2ⁿ

But what if you want to sum from any arbitrary k, like 4? I've tried looking at Pascal's triangle for nice values of n and k, but haven't been able to see a pattern. I would really appreciate any help with this. I want to apply this to combinations for large n, which are impractical to compute.

Thank you in advance.

 

[Stephen Tashi]

I don't know any nice formula for [itex] \sum_{k=0}^m \binom{n}{k} [/itex] Your question made me curious and I searched the web. It apparently doesn't know a nice formula either. Perhaps if you give an example of the kind of computation you are trying to do, someone will see a way to compute the result - at least compute it on a computer.

 

[bpet]

The sum could be expressed in terms of the incomplete beta function, e.g. using the cdf of the binomial distribution with p=1/2.

 

[mXSCNT]

For large n the binomial distribution is approximated by a normal distribution, so if you only want a close approximation you could use that.

 

[blue_raver22]

I understand your frustration in trying to find a formula for the sum of combinations from any arbitrary k to n. This is a common problem in mathematics and often requires creative thinking and problem-solving skills to find a solution.

One approach you could try is to use the binomial theorem, which states that for any non-negative integer n, the sum of the coefficients of the terms in the expansion of (x+y)^n is equal to 2^n. This may not directly apply to your specific problem, but it could provide some insight and help you find a pattern.

Another approach could be to use combinatorial identities, such as the Vandermonde's identity, which states that the sum of combinations of k from 0 to m and n-k from 0 to m is equal to the combination of n from 0 to 2m. This may also provide some clues and help you find a formula for the sum of combinations from any arbitrary k to n.

Additionally, you could consider using computer programs or algorithms to help you calculate combinations for large values of n. This can save time and effort, and also allow you to focus on finding a formula for the sum of combinations rather than computing individual values.

I hope these suggestions are helpful and wish you the best of luck in finding a solution to your problem. Remember, as a scientist, it is important to keep exploring and trying different approaches until you find a satisfactory solution. Keep up the curiosity and determination in your research!