SOLUTION: In the design of a certain mechanical product, ten components are to be stacked into a cylindrical casing, in a manner that minimizes the impact of shocks. One end of the casing is

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Question 1098386: In the design of a certain mechanical product, ten components are to be stacked into a cylindrical casing, in a manner that minimizes the impact of shocks. One end of the casing is designated as the top and the other end is the bottom.
a. If all components are different, how many different stacks are possible?
b. If six components are identical to one another, but the others are different, how many different stacks are possible?
c. If four components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible?
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
In the design of a certain mechanical product, ten components are to be stacked into a cylindrical casing, in a manner
that minimizes the impact of shocks. One end of the casing is designated as the top and the other end is the bottom. 

a.  If all components are different, how many different stacks are possible?

    10! = 10*9*8*7*6*5*4*3*2*1 = 3628800  different stacks.



b.  If six components are identical to one another, but the others are different, how many different stacks are possible?

     = 10*9*8*7 = 5040  different stacks.


c.  If four components are of one type and identical to one another, and four components are of another type 
    and identical to one another, but the others are different, how many different designs are possible?

     = 6300  different stacks.


Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
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 .

b. If 6 of the 10 pieces are identical, many of those
assemblies will look exactly the same. They only differ in the order those 6 identical pieces were added.
There are 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 different
designs.

c. If 4 pieces are of kind A, another 4 pieces are of kind B,
and the remaining pieces are different from all others,
there when will be 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 identical-looking design that differ only in the order the B pieces were arranged.
All in all, there are
different-looking designs.
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