How many 3 digit positive integers have exactly 1 even digit?
(A) 350 (B) 450 (C) 375 (D) 75 (E) 125
Well, we know that we can only use the numbers between 100 and 999. We also know that 0 cannot be used as the first number because that would result in a 2 digit number. The easiest was to find the answer is to write out the possibilities.
The first group is going to be the integers that start with an even number.
First number: 2, 4, 6, 8
Second number: 1, 3, 5, 7, 9
Third number: 1, 3, 5, 7, 9
This means that the number of possibilities equal 4 x 5 x 5 which equals 100.
The second group is going to be integers in which the second number is even.
First number: 1, 3, 5, 7, 9
Second number: 0, 2, 4, 6, 8
Third number: 1, 3, 5, 7, 9
This means that the number of possibilities equal 5 x 5 x 5 which equals 125.
The second group is going to be integers in which the third number is even.
First number: 1, 3, 5, 7, 9
Second number: 1, 3, 5, 7, 9
Third number: 0, 2, 4, 6, 8
This means that the number of possibilities equal 5 x 5 x 5 which equals 125.
Now we just add the numbers to find the answer
100 + 125 + 125 = 350
The answer to this equation is (A) 350.
At first I had no idea how to approach this question. The first thing I thought of was to list all the numbers but that would take a very long time to do and I thought that there must be an easier, more organized way to do it (I remembered doing something like this in elementary school when they gave us a few numbers and we were supposed to find out how many combinations there were). So I thought to categorize them by which number was even and then writing all the possibilities and multiplying them. This quickly gave me the answer in an organized fashion that's easy to understand.
In the process of problem solving I learned that the way I first think about things may not always be the best way to approach a question. I should always keep my mind open to other solutions to each question.
