Problem Solving

Of the three digit integers greater than 700, how many two digits that are equal to each other and the remaining digit different from the other two?

a) 90
b) 82
c) 80
d) 36
e) 45

Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

Integers with exactly 2 digits the same = Total integers - Integers with all 3 digits the same - Integers with all 3 digits different.

Total integers:
To count consecutive integers, use the following formula:
Number of integers = biggest - smallest + 1.
Thus:
Total = 999 - 701 + 1 = 299.

Integers with all 3 digits the same:
777, 888, 999.
Number of options = 3.

Integers with all 3 digits different:
Number of options for the hundreds digit = 3. (7, 8, or 9)
Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.)
Number of options for the units digit = 8. (Any digit 0-9 other than the two digits already used.)
To combine these options, we multiply:
3*9*8 = 216.

Thus:
Integers with exactly 2 digits the same = 299-3-216 = 80.

The correct answer is C.

nchaswal wrote:Of the three digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

a) 90
b) 82
c) 80
d) 36
e) 45

Here's an alternate method.

The three digit integers greater than 700 are the integers from 701 - 999 inclusive.

Start with the integers that begin with 7.

Any integer that ends in 7 except 777 will work.

So we have 707, 717, 727 ... 797 for a total of 9.

Also integers except 777 that have 7 as the second number will work. 770, 771, 772, 773 ... 779. For a total of 9.

Then other than 777 any integer with the second two digits the same will work.

711, 722, 733 ... 799, for a total of 8.

So we have 9 + 9 + 8 = 26.

This pattern holds for the integers from 801 to 899, 9 + 9 + 8 = 26, and for the integers from 901 to 999, 9 + 9 + 8 = 26.

So, so far we have 3 x 26 = 78 integers that fit.

(If you were tempted to go with 78 as the answer, you would be saved by the fact that 78 is not one of the answer choices.)

800 and 900 work too.

So 78 + 2 = 80.

The correct answer is C.