A formula to calculate summation of nCr with r ranging from a to b(b>a) I want to find a simple formula for this 
$$\sum_{r=a}^b nCr  =  ?$$
for example 
$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$
Here a and b are natural numbers and r is being incremented by 2 after every step. so if (a,b) = (1,7) then r will have values 1,3,5 and 7. n will also be a natural number which will be a constant.
I searched the net but i could only find it for r ranging from 0 to n.
 A: The formula ncr of combinatorics can be expressed in summation form or as a series by using MULTIPLE SIGMA notation. Look up the link I have posted here and you will find a paper titled-"THE FORMULA nCr REVISITED" by Soumendra Nath Banerjee.
The proof of the formula is also given in the paper itself.
The formula is as below-------
How the formula looks
Open the link below and scroll down till you find the paper titled "THE FORMULA nCr REVISITED"
THE FORMULA nCr REVISITED
I am also giving the abstract below.
I hope this answers your question. Thank you for your patient reading.
Abstract-
A formula expressing n C r in summation form is formulated by the use of algorithmic counting techniques. Initially, a general counting problem is mathematically modeled and its solution is given by a formula derived using algorithmic counting. Thus, by generalization a formula for nCr as a series is obtained.
Keywords: n C r, summation, algorithm, counting, mathematical modeling, generalization, series.
