to calculate the probability, you have to think about all the possible ways he could have gotten the 3 pieces in 6 or fewer kills. there's 15 such combinations. they are:

kill 4 top route, get drop
kill 1 middle route, get drop
kill 1 bottom route, get drop

kill 3 top route, get drop
kill 1 middle route, get drop
kill 2 bottom route, get drop

following that pattern:

3
2
1

2
1
3

2
2
2

2
3
1

1
1
4

1
4
1

1
2
3

1
3
2

3
1
1

2
1
2

2
2
1

1
1
2

1
2
1

1
1
1

so then you have to calculate the probability for each of these cases. doing that is very simple. for the first case, it is

[1-(9/10)^4]*(1/10)*(1/10)=0.00349

the second case is
[1-(9/10)^3]*(1/10)*[1-(9/10)^2]=0.005149

the third case comes out to be the same as case 2, so another 0.005149

case 4 = 0.005149

case 5 = 0.006859

case 6 = 0.005149

case 7 = 0.00349

case 8 = 0.00349

case 9 = 0.005149

case 10 = 0.005149

so these are the cases for exactly 6 skills. thus, the probability of getting 3 pieces in exactly six kills is the sum of these, or 0.048223.

I don't feel like doing the rest of the calculations, but you get the idea.