GeminiHeat wrote:
Of the three-digit positive integers whose three digits are all different and nonzero, how many are odd integers greater than 700?
(A) 84
(B) 91
(C) 100
(D) 105
(E) 243
We know the integers must be greater than 700 and they must be odd; we can divide them into 6 groups: 7EO, 7OO, 8EO, 8OO, 9EO and 9OO where E denotes an even digit and O an odd digit. Let’s determine the number of integers in each group.
7EO:
The hundreds digit is 7, so it’s only 1 choice. Since all the digits are nonzero, E can be 4 numbers: 2, 4, 6 or 8. Since the digits are all different, O can be 4 numbers: 1, 3, 5 or 9. Therefore, there are 1 x 4 x 4 = 16 such integers.
7OO:
Since all the digits are different, O can only be 4 numbers: 1, 3, 5 or 9. However, the second O can only be 3 numbers since it cannot be the same as the first O. Therefore, there are 1 x 4 x 3 = 12 such integers.
8EO:
The hundreds digit is 8, so there is only 1 option. Since all the digits are different and nonzero, E can only be 3 numbers: 2, 4, or 6. Since the last digit is O, an odd digit, O can any of the 5 odd digits: 1, 3, 5, 7 or 9. Therefore, there are 1 x 3 x 5 = 15 such integers.
8OO:
The first O can be any of the 5 odd digits: 1, 3, 5, 7 or 9. However, the second O can only be 4 numbers since it cannot be the same as the first O. Therefore, there are 1 x 5 x 4 = 20 such integers.
9EO:
This is analogous to 7EO, so there should be 16 such integers.
9OO:
This is analogous to 9EO, so there should be 12 such integers.
Therefore, there are 16 + 12 + 15 + 20 + 16 + 12 = 91 odd integers that are greater than 700 where all digits are different and nonzero.
Answer: B