Max@Math Revolution wrote:[GMAT math practice question]
How many different three-digit numbers can be formed which contain two digits that are the same, and a third digit that is different from the other two?
A. 196
B. 216
C. 243
D. 256
E. 316
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.
Biggest 3-digit integer = 999.
Smallest 3-digit integer = 100.
Thus:
Total = 999 - 100 + 1 = 900.
Integers with all 3 digits the same:
111, 222, 333, 444, 555, 666, 777, 888, 999.
Number of options = 9.
Integers with all 3 digits different:
Number of options for the hundreds digit = 9. (Any digit 1-9)
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:
9*9*8 = 648.
Thus:
Integers with exactly 2 digits the same = 900 - 9 - 648 = 243.
The correct answer is
C.
Alternate approach:
Case 1: The hundreds and the tens digits are the same, the units digit is different
Number of options for the hundreds digit = 9. (Any digit 1-9.)
Number of options for the tens digit = 1. (Must be the same as the hundreds digit.)
Number of options for the units digit = 9. (Any digit 0-9 other than the digit already used.)
To combine these options, we multiply:
9*1*9 = 81.
Case 2: The hundreds and the units digits are the same, the units digit is different
Number of options for the hundreds digit = 9. (Any digit 1-9.)
Number of options for the units digit = 1. (Must be the same as the hundreds digit.)
Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.)
To combine these options, we multiply:
9*1*9 = 81.
Case 3: The tens and the units digits are the same, the hundreds digit is different
Number of options for the hundreds digit = 9. (Any digit 1-9.)
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 = 1. (Must be the same as the tens digit)
To combine these options, we multiply:
9*9*1 = 81.
Total options = Case 1 + Case 2 + Case 3 = 81 + 81 + 81 = 243.
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