How can I calculate this summation? $\sum_{x=60}^{100} {100\choose x} $?

Question

How can I calculate this summation?
$$\sum_{x=60}^{100} {100\choose x} $$ ?

I don't have idea how to calculate it, I tried to arrive at a probability expression of a random variable that is binomial ($Bin(n,p)$) but But I did not succeed.

Answer 1

Hint:

The following is standard: $$\sum_{k=0}^n\binom nk=2^n.$$

Answer 2

There is probably no way to avoid summing at least few of these (rather large) binomial coefficients, but at least you can limit number of those if you utilize hint below.

Hint:

$$ \sum_{x=0}^{100} {100\choose x} = 2\sum_{x=50}^{100} {100\choose x}-\binom{100}{50} $$

Answer 3

$$ \sum_{x=0}^{100} {100 \choose x} = {(2)}^{100} $$ And using the fact that $ {n \choose r} = {n \choose n-r} $ it can be shown that $$ \sum_{x=50}^{100} {100 \choose x} = {(2)}^{99} $$ So $$ \sum_{x=60}^{100} {100 \choose x} = {(2)}^{99} - \sum_{x=50}^{59} {100 \choose x} $$