SOLUTION: Find the number of 10-digit numbers whose product of digits is 8

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Question 546015: Find the number of 10-digit numbers whose product of digits is 8
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
To have product 8 the 10 digits have to be one of these types:

A.  those with this group of digits:                  2,2,2,1,1,1,1,1,1,1

B.  those with this group of digits with 2 left of 4: 2,4,1,1,1,1,1,1,1,1

C.  those with this group of digits with 4 left of 2: 4,2,1,1,1,1,1,1,1,1

D.  those with this group of digits:                  8,1,1,1,1,1,1,1,1,1

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There are 10 positions to choose in a 10-digit number

We calculate the number of 10-digit numbers of type A:
The 3 2's can be placed in 10C3 or 120 positions

We calculate the number of 10-digit numbers of type B
The 2 and 4 with 2 left of 4 can be placed in 10C2 or 45 positions

We calculate the number of 10-digit numbers of type C
The 4 and 2 with 4 left of 2 can be placed in 10C2 or 45 positions

We calculate the number of 10-digit numbers of type D
The 8 can be placed in 10C1 or 10 positions
 
That's 120+45+45+10 = 220 ways

Edwin