Math Help - volume of part of simplex

1. volume of part of simplex

We have a regular simplex x1+...xn=1. The formula to calculate its volume is well-known.
I am interested to find a volume of part of the simplex, which satisfies x1+...+xk<A, 0<A<1,k<n
Thank you in advance

2. That condition is subtle--I wrote up two separate replies about how it doesn't really make sense, and each time I realized I was interpreting it wrong. My first instinct is to try subdivision. If I come up with anything more specific I'll post back. Interesting problem...