Efficient sampling from a 'partial' binomial distribution

I am want to sample from the binomial distribution B(n,p) but with an additional constraint that the sampled value belongs in the range [a,b] (instead of the normal 0 to n range). In other words, I have to sample a value from binomial distribution given that it lies in the range [a,b]. Mathematically, I can write the pmf of this distribution (f(x)) in terms of the pmf of binomial distribution bin(x) = [(nCx)*(p)^x*(1-p)^(n-x)] as

sum = 0
for i in range(a,b+1):
    sum += bin(i)

f(x) = bin(x)/sum

One way of sampling from this distribution is to sample a uniformly distributed number and apply the inverse of the CDF(obtained using the pmf). However, I don't think this is a good idea as the pmf calculation would easily get very time-consuming.

The values of n,x,a,b are quite large in my case and this way of computing pmf and then using a uniform random variable to generate the sample seems extremely inefficient due to the factorial terms in nCx.

What's a nice/efficient way to achieve this?