2074. Ant Counting

Bessie was poking around the ant hill one day watching the ants march to and fro while gathering food. She realized that many of the ants were siblings, indistinguishable from one another. She also realized the sometimes only one ant would go for food, sometimes a few, and sometimes all of them. This made for a large number of different sets of ants!

Being a bit mathematical, Bessie started wondering. Bessie noted that the hive has $T$ $(1\le T\le 1000)$ families of ants which she labeled $1$..$T$ ($A$ ants altogether). Each family had some number $N_i$ $(1\le N_i\le 100)$ of ants.

How many groups of sizes $S,S+1,\dots ,B$ $(1\le S\le B\le A)$ can be formed?

While observing one group, the set of three ant families was seen as $1,1,2,2,3$, though rarely in that order. The possible sets of marching ants were:

• 3 sets with 1 ant: {1} {2} {3}
• 5 sets with 2 ants: {1,1} {1,2} {1,3} {2,2} {2,3}
• 5 sets with 3 ants: {1,1,2} {1,1,3} {1,2,2} {1,2,3} {2,2,3}
• 3 sets with 4 ants: {1,2,2,3} {1,1,2,2} {1,1,2,3}
• 1 set with 5 ants: {1,1,2,2,3}

Your job is to count the number of possible sets of ants given the data above.