Question 1098386
a. There are 10 choices for bottom piece.
For each of those choices there are 9 for the piece to put on top of the first, and so on.
The possibilities literally multiply.
The total is {{{10!=10*9*8*"..."*3*2*1=highlight(3628800)}}} .
 
b. If 6 of the 10 pieces are identical, many of those {{{3628800}}}
assemblies will look exactly the same. They only differ in the order those 6 identical pieces were added.
There are {{{6!=6*5*4*3*2*1=720}}} different orders,
but if you cannot tell those 6 pieces apart,
we count of just one design
each set of 720 assemblies that differ only in the order the identical pieces were arranged.
Then, there are {{{10!/6!=10*9*8*7=3628800/720=highlight(5040)}}} different 
designs.
 
c. If 4 pieces are of kind A, another 4 pieces are of kind B,
and the remaining {{{10-4-4=2}}} pieces are different from all others,
there when will be {{{4!=4*3*2*1=24}}} ways for the same design
that differ only in the order the A pieces appear. And even when the orderof the A pieces is the same, there will be {{{24}}} identical-looking design that differ only in the order the B pieces were arranged.
All in all, there are
{{{10!/(4!*4!)=3628800/(24*24)=highlight(6300)}}} different-looking designs.