Hello, 

This seems to involve Rencontres numbers $D_{n,r}$, the number of permutations in symmetric group $S_n$ with $r$ fixed points, for example, see <a href="https://oeis.org/wiki/Rencontres_numbers">OEIS Rencontres numbers</a> or this post, <a href="http://terrytao.wordpress.com/2011/11/23/the-number-of-cycles-in-a-random-permutation/">The number of cycles in a random permutation</a>, in the blog of Professor Tao.

Let $P_n$ be the set of partitions of $10$ into $n$ (not necessarily distinct and not necessarily non-zero) components. I.e, 

$$
P_n = \{(a_1,a_2,\dots,a_n): \sum_{i}a_i = 10, 0 \leq a_i \leq 10\}.
$$

Then the probability that $T=n$ is

$$
P(T=n) = \sum_{\pi = (a_i)\in P_n}\left(\frac{D_{10,a_1}}{10!}\right)\left(\frac{D_{10-a_1,a_2}}{(10-a_1)!}\right)\dots\left(\frac{D_{10-a_1-\dots-a_{n-1},a_n}}{(10-\sum_{i=1}^{n-1}a_i)!}\right).
$$