Bunuel
How many four-digit numbers can you form using ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if the numbers can be used only once?
A. 5040
B. 4536
C. 4050
D. 4036
E. 504
Solution:
If the number does not have a zero digit, then we have 9P4 = 9 x 8 x 7 x 6 = 3024 such numbers.
If the number has a zero digit and the zero digit is the units digit, then we have 9P3 = 9 x 8 x 7 = 504 ways to permute the other 3 digits. Therefore, we have 504 such numbers. Similarly, if the zero digit is the tens digit or the hundreds digit, we would also have 504 numbers in each of these cases. Of course, we can’t have the zero digit as the thousands digit since a 4-digit number can’t begin with 0.
Therefore, there are a total of 3024 + 504 x 3 = 4,536 such numbers.
Alternate Solution:For the thousands digit, there are 9 choices since 0 cannot be the thousands digit of a four-digit integer. For the hundreds digit, there are also 9 choices since we exclude the number in the thousands digit but include 0. For the tens and units digit, there are 8 and 7 choices, respectively. Thus, there are 9 x 9 x 8 x 7 = 4,536 such numbers.
Answer: B