SOLUTION: How many four digit numbers can be formed from 0-9 if: repetitions are allowed the number is less than 5000 and repetitions are allowed? the number is less than 5000 and no repe

Two ways to do that one.

First way:

Choose the 1st digit any of 9 ways.  (can't use 0)
Choose the 2nd digit any of 10 ways, (can use 0)
Choose the 3rd digit any of 10 ways, (can use 0)
Choose the 4th digit any of 10 ways, (can use 0)

9×10×10×10 = 9000

Second way: 

There are 9999 positive integers from 1 thru 9999 inclusively.
Of those the 999 integers from 1 through 999 inclusively are 
not four-digit numbers.  So there are 9999-999 = 9000 

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Choose the 1st digit any of 4 ways, (1,2,3,or 4)
Choose the 2nd digit any of 10 ways.
Choose the 3rd digit any of 10 ways.
Choose the 4th digit any of 10 ways.

Answer: 4×10×10×10 = 4×103 = 4000

Second way: 

There are 4999 positive integers less than 5000.
Of those the 999 integers from 1 through 999 inclusively are 
not four-digit numbers.  So there are 4999-999 = 4000 

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Choose the first digit first as that is the most restrictive.

Choose the 1st digit any of the 4 ways (as 1,2,3,or 4
Choose the 2nd digit in any of the 9 remaining ways.
Choose the 3rd digit in any of the 8 remaining ways.
Choose the 4th digit in any of the 7 remaining ways.

Answer:  4×9×8×7 = 2016 ways 

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First way:

Choose the 4th digit only 1 way, as 5.
Choose the 1st digit any of 9 ways.  (can't use 0)
Choose the 2nd digit any of 10 ways, (can use 0)
Choose the 3rd digit any of 10 ways, (can use 0)


1×9×10×10 = 900

Second way:

Choose the 3-digit number that precedes the 5. 

There are 999 positive integers from 1 thru 999 inclusively.
Of those the 999 integers from 1 through 99 inclusively are 
not three-digit numbers.  So there are 999-99 = 900 3-digit
numbers that you can put a 5 on the right end as a 4th digit
ending in 5.  Answer: 900

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Choose the last digit first as that is the most restrictive.

Choose the 4th digit only 1 way, as 5.
Choose the 1st digit any of 8 remaining ways.  (can't use 0 or 5)
Choose the 2nd digit any of 8 remaining ways, (can use 0, but not 5)
Choose the 3rd digit any of 7 ways.

1×8×8×7

Answer:  1×8×8×7 = 448 ways 

Edwin