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# Counting problem involving sets

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A set of five positive integers has a mean, median, and range of 7. How many distinct sets whose members are listed from least to greatest could have these properties? Note: {4, 6, 7, 7, 11} is one such set to include.

I know that 7 must be in the set, and it is in the middle. Also, the sum of the other numbers must be 28. However, I cannot think of a good way to do this without just brute forcing every single value. Could somebody help me with this? Thanks!

May 24, 2024

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To find the possible sets, let's call the integers a, b, c, d, and e, listed from least to greatest. We know that the mean, median, and range are all 7.

The mean of the set is the sum of the numbers divided by their count. So, we have a + b + c + d + e = 5 * 7 = 35.

Since the median is the middle number when the set is sorted, the third number must be 7. So, we have a, b, 7, d, and e.

The range is the difference between the greatest and least numbers. Since the range is 7, we have e - a = 7.

Now, since e > a, we know that e is at least 8. This means the maximum value for e is 15, since e + 7 = a + 14.

To find the number of distant sets, we need to consider all possible values for e.

If e = 8, then a = 1. The set would be {1, b, 7, d, 8}.

If e = 9, then a = 2. The set would be {2, b, 7, d, 9}.

Continuing this pattern, if e = 15, then a = 8. The set would be {8, b, 7, d, 15}.

So, we have a total of 15 - 8 + 1 = 8 distant sets whose members are listed from least to greatest and have the given properties.

May 24, 2024