How many 4-digit whole numbers using the digits in the set {1,2,3,4,5,6,7,8,9,0}can be formed using the restrictions below.
For purposes of this problem, the whole number can be "padded" with zeros on the left. So a 0014 is a good number,(IE numbers can start with zeros).
Each part below is is independant of the previous part.
a. Numbers do not contain the same digit twice, (without replacement)
Examples would be 0692, 8591
Choose digit #1 10 ways, and then choose digit #2 9 ways,
and then choose digit #3 8 ways, and then choose digit #4
7 ways.There are 10×9×8×7 = 5040 ways.
b. Numbers end with an even digit and can repeat digits, (with replacement)
Examples would be 8736, 4994, 7770, 0000, 2010
We do not have to choose the digits in any certain order.
So we will choose digit #4, then digit #1, then digit #2,
and finally digit #3.
Choose digit #4 5 ways (2,4,6,8, or 0), and then choose
digit #1 10 ways, and then choose digit #2 10 ways, and
then choose digit #3 10 ways. There are
5×10×10×10 = 5000 ways.
c. Numbers begin with an odd digit and can repeat digits,
(with replacement)
Examples: 1887, 9742, 5555
Choose digit #1 5 ways (1,3,5,7,9), and then choose
digit #2 10 ways, and then choose digit #3 10 ways, and
then choose digit #4 10 ways. There are
5×10×10×10 = 5000 ways.
d. Numbers begin and end with a 1, (with replacement)
Examples: 1871, 1661, 1111
Choose digit #1 1 way, and then choose digit #2 10 ways,
and then choose digit #3 10 ways, and then choose digit
#4 1 way. There are 1×10×10×1 = 100 ways.
e. Numbers have exactly three digits which are 7s (with replacement)
Examples: 7771, 7707, 7477, 0777
There are three 7's and one non-7. We choose the position
for the non-7 any of 4 ways, and then the digit for the non-7
9 ways, then 1 way each for the 7's.
There are 4×9×1×1 = 36 ways.
Edwin