I will attempt to count them too ...
I know that six of the rumbers are
1,1,3,3,5,5, and I am assuming that the others four must be 2,4 or 6 each.
2 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 5 | 5 | 10!/(4!*2*2*2)=10!/(4!*8)=18900 | |
2 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 5 | 5 | 10!/(3!*8)=75600 | |
2 | 2 | 2 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 75600 | |
2 | 2 | 4 | 4 | 1 | 1 | 3 | 3 | 5 | 5 | 10!/(2*2*8)=113400 | |
2 | 2 | 4 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 10!/(2*8)=226800 | |
2 | 2 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 113400 | |
2 | 4 | 4 | 4 | 1 | 1 | 3 | 3 | 5 | 5 | 75600 | |
2 | 4 | 4 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 226800 | |
2 | 4 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 226800 | |
2 | 6 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 75600 | |
4 | 4 | 4 | 4 | 1 | 1 | 3 | 3 | 5 | 5 | 18900 | |
4 | 4 | 4 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 75600 | |
4 | 4 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 113400 | |
4 | 6 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 75600 | |
6 | 6 | 6 | 6 | 1 | 1 | 3 | 3 | 5 | 5 | 18900 | |
3*18900+6*75600+3*113400+3*226800 = 1530900 |
So I get the probability as \(\frac{1530900}{6^{10} }= \frac{175}{6912} \approx 0.025318287037037\) Note: this has been edited (1st edit)
That is if i have made no mistakes.
There would be a number of more sensible ways to do this.