How to find a sum of the first r binomial coefficients for fixed n?

Question

I have already tried the basic way to solve this series but it takes time for the larger values of n & r. Is there any way to reduce this expression in a single expression whose time complexity doesn't depend on the value of n OR r.Range r,n<=10^5

NOTE: here r < n i.e. i have to find the sum of first r+1 terms of this series.

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I have already read this question but it doesn't help me:

Algorithm to find Sum of the first r binomial coefficients for fixed n modulo m

Answer 1

AFAIK, there is no such expression to which it can be reduced. But it can be done in O(r) time complexity as follows.

Consider an array A, where A[i] stores ⁿc_i. Then we can easily verify that A[i] = A[i-1].(n-i+1)/(i)

So

A[0] = 1;
for(int i=1;i<=r;i++){
    A[i] = A[i-1].(n-i+1)/(i);
}
int ans = 0; //The required answer
for(int i=0;i<=r;i++){
    ans = ans+A[i];
}