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How many five-digit numbers may be made from the five digits 1, 2, 3, 4, 5.
If a digit may be used exactly once and neither of the first two digits can be 5?
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In all, you have 5 digits in your possession.
As 1st digit of your number, you may have any of 4 digits {1,2,3,4}. 4 options.
As 2nd digit of your number, you may have any of 3 remaining digits. 3 options
As 3rd digit of your number, you can use any of remaining 5-2 = 3 digits. 3 options.
As 4th digit of your number, you can use any of remaining 5-3 = 2 digits. 2 options.
As 5th digit of your number, you can use any of remaining 5-4 = 1 digits. 1 options.
So, the answer to the problem's question is 4*3*3*2*1 = 72 possible 5-digit numbers under the imposed conditions.
Solved and carefully/thoroughly explained.