Ants assembly

The probability of exactly $0 \leq n \leq 6$ ants meeting at the center is

$p(n) = \left( \frac 1 3 \right) ^n \cdot \left( \frac 2 3 \right) ^{6-n} \cdot \binom 6 n$

This is because the probability of exactly $n$ specific ants moving to the center is $\left( \frac 13 \right) ^n$ , the probability of the other $6-n$ ants moving somewhere else is $\left( \frac 23 \right) ^{6-n}$ , and finally there are $\binom 6n$ ways to choose $n$ specific ants.

Now, we only have to add up all probabilities

$p(3)+p(4)+p(5)+p(6) = \frac {233}{729}$

and so the answer is $233+729 = \boxed{962}$ .