Combinatorics

Say we have a string of the form AABCCC where A is a letter, C is a digit, B can be either and there are no repeated letters or digits.

Can somebody explain why the number of permutations is equal to:

26 * 25 * (24 + 7) * 10 * 9 * 8

and not:

26 * 25 * (24 + 10) * 9 * 8 * 7

Break up the ()s.
The 1st = (26*25*[24]*10*9*8) + (26*25*[7]*10*9*8)
The 2nd = (26*25*[24]*9*8*7) + (26*25*[10]*9*8*7)

The right terms are equal, and correctly account for the case where B is a #.

The left terms are not equal, and the wrong one incorrectly restricts the Cs, since now B is a letter.

I believe it's because the second one gives preference to B's being a number though it's not necessarily true. Doing so unnecessarily gets rid of possibilities for C.

Thanks. That makes much more sense.

This is one of the best threads I have ever seen on /sci/, just fyi.

Another way to think about the 1st eq. is to look at the two cases:
B is a letter: 26*25*24*10*9*8     (1)
B is a digit : 26*25*10*9 *8*7     (2)
             = 26*25*7 *10*9*8     (3)
(1)+(3) yields 1st eq.

on the other hand, using
               26*25*24*9 *8*7     (1x)
(1x)+(2) yields 2nd eq.; you can see in the case "B is a letter", this is equivalent to excluding a digit for the C's.

(actually this is the same as
>>2
with expanded explanation)

A general principle that is helpful is to specify the least restricted positions(s) last.  The third position is the least restricted.  Once you get to it, you have already used up two letters and three digits for the other five positions, leaving the third position with 31 characters available for it.