Question number 28. Consider the set of all the seven-digit numbers that can be obtained using all the digits 1, 2, 3, 4, 5, 6, 7 for each number. List the numbers of the set in increasing order and divide the list exactly in the middle into two parts of the same length. What is the last number of the first half of the list? So let's write down some of the numbers in this list. We start with the smallest number so that's 1, 2, 3, 4, 5, 6, 7 and the end here would be that number in reverse 7, 6, 5, 4, 3, 2 and 1. That's the largest number possible and here in the middle we have a number that we have to find and we're not just looking for the number in the middle, we're looking for the last number in the first half of the list but the average here is the number 4, 4, 4, 4, 4, 4 so we have exactly 7 fourths. Now even though it is not possible to obtain this number because we're using one digit exactly once, let's keep in mind that if we had all digits available this would be their average and we'll need that in a moment. So we see now that the question here cannot be answered with A because that's the smallest number and what should the middle look like? Well, if we take the numbers here in the middle, we see that cutting the list in half gives us an exact value of 4 so somewhere we should start with a 4 here in the middle and now removing 4 from the list, we have used that already, we have the numbers 3 and 5 next to each other like that so which one should we decide on? Well, we don't have the choice of 5 in our answers here so that should be easy. We have a 4, 3 here in D and we have a 4, 3 in E so right away we have to discount B and C also cannot be the answer because the next number after 4 needs to be a 3 or a 5. So then with a 4, 3 to begin we should check which number is closer to the middle. Is it 4, 3, 5, 2, 6, 1, 7 or is it 4, 3, 7, 6, 5, 2, 1? And we also notice that this number here is the 7, 6, 5, 2, 1 is the largest of the possible numbers using the remaining digits and closer to the average then 4, 3, 5, 2, 6, 1, 7 so we should choose that as our answer it is a lot closer to the middle here to the average of the smallest and the largest number and the 7, 6, 5, 2, 1 make this the number in the middle here that's as large as possible so it has to be the end number and in the first half and we choose that as our answer here and that is E.

seven-digit numbers

digits 1 to 7

increasing order

first half

last number