Help: sum of binomial coefficents

[thealchemist83]

Help: sum of binomial coefficents !

Hello!
I cannot figure out how to derive a closed formula for the sum of "the first s" binomial coefficients:

$$\sum_{k=0}^{s} \left({{n}\atop{k}}\right)$$

with $$s<n$$

Could you please help me find out some trick to derive the formula... I've an exam on monday!

Thank you very much!

 

[mathman]

Hello!
I cannot figure out how to derive a closed formula for the sum of "the first s" binomial coefficients:

$$\sum_{k=0}^{s} \left({{n}\atop{k}}\right)$$

with $$s<n$$

Could you please help me find out some trick to derive the formula... I've an exam on monday!

Thank you very much!

I don't believe there is any such formula.

 

[sutupidmath]

well, i think there is one, because as long as i remember i have seen it in a textbook, but it is quite long i think, and i cannot remember how it was right now. I am going to look at it.

 

[atqamar]

I remember that the n-th binomial coefficients can be seen on the n-th line of the Pascal's Triangle. I also remember that the sum of the numbers in the n-th line of the Pascal's Triangle is $$2^n$$. NOTE: The first row is the 0-th row, and the next line is the 1-st.

Edit: Here is some additional information: http://en.wikipedia.org/wiki/Binomial_coefficient#Formulas_involving_binomial_coefficients .

 

[mathman]

I remember that the n-th binomial coefficients can be seen on the n-th line of the Pascal's Triangle. I also remember that the sum of the numbers in the n-th line of the Pascal's Triangle is $$2^n$$. NOTE: The first row is the 0-th row, and the next line is the 1-st.

Edit: Here is some additional information: http://en.wikipedia.org/wiki/Binomial_coefficient#Formulas_involving_binomial_coefficients .

Your answer 2ⁿ is for s=n, the original question was for s<n.

 

[Count Iblis]

No formula exists in this case. See the book A=B for how to simplify binomial summations. There exists simple algorithms that will yield a formula or will tell you that no formula exists.