If you think logically , the counting can be done in less than 1 minute but of course it requires practise.
$First Observation:$ There will be at least one of the $4,5,6$ because even if you take $3$ three times, you cannot make a $10$.
Number of ways in which 4 can be involved:
$4,5,1$ total $3! = 6$ ways
$4,4,2$ total $3$ ways
$4,3,3$ total $3$ ways
Number of ways in which 5 can be involved:
$5,4,1$ has a $4$ so dont count
$5,3,2$ total $3! = 6$ ways
Number of ways in which 6 can be involved:
$6,3,1$ total $3! = 6$ ways
$6,2,2$ total $3$ ways
$Total count = 27$. Probability is $1/8$