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Of the three-digit positive integers that have no digits equal to zero
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12 Oct 2021, 09:55
12 Oct 2021, 09:55
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Of the three-digit positive integers that have no digits equal to zero, how many have two digits that are equal to each other and the remaining digit different from the other two?
A. 24
B. 36
C. 72
D. 144
E. 216
A. 24
B. 36
C. 72
D. 144
E. 216
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Re: Of the three-digit positive integers that have no digits equal to zero
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12 Oct 2021, 10:03
12 Oct 2021, 10:03
Carcass wrote:
Of the three-digit positive integers that have no digits equal to zero, how many have two digits that are equal to each other and the remaining digit different from the other two?
A. 24
B. 36
C. 72
D. 144
E. 216
A. 24
B. 36
C. 72
D. 144
E. 216
Take the task of creating suitable 3-digit numbers and break it into stages.
Stage 1: Select the single digit that will be different from the other 2 digits
We can choose any of the following 9 digits: 1, 2, 3, 4, 5, 6, 7, 8 or 9
So, we can complete stage 1 in 9 ways
Stage 2: Choose where that single digit (selected above) will be placed
There are 3 places (hundreds position, tens position or units position) where we can place this digit.
So, we can complete stage 2 in 3 ways.
At this point, we have 2 spaces left, and these spaces will be filled by the same digit.
Stage 3: Select the digit to occupy those two remaining spaces
There are 8 remaining digits from which to choose, so we can complete this stage in 8 ways.
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create the required 3-digit number) in (9)(3)(8) ways (= 216 ways)
Answer: E
Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.
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Re: Of the three-digit positive integers that have no digits equal to zero [#permalink]
12 Oct 2021, 10:03
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