Seven-element lists of integers with conditions on means

Question

Can anyone help me with this? It seems that I can't assume the seven elements to be $x,x+1,\dots,x+6$. I feel something is wrong with this problem.

How many ways are there to write seven-element data lists of integers in numerical order to fit the following criteria: the mean is 20; the median is 21; the mode is 22 and unique; and the range is 10?

Answer 1

21, being the median, is the fourth element in the ordered list; there are three elements above and below it. 22, being the mode and greater than 21, can only appear two or three times.

If 22 appears three times we have the following list:

12 __ __ 21 22 22 22

The remaining two slots must sum to 41 and be less than or equal to 21. Only (20, 21) fits.

If 22 appears twice, it cannot be the greatest number since that would leave the fifth number in the list without a suitable assignment (either a third 22, or 22 would not be a unique mode). Thus we must have the following:

__ __ __ 21 22 22 __

All remaining numbers must now be distinct so that 22 may remain a unique mode. Try 23 as the largest number:

13 __ __ 21 22 22 23

The last two empty slots must sum to 39 and be less than 21; only (19, 20) fits. We can continue incrementing the largest number (and the smallest, since the range is 10) and see what comes of the second and third slots:

• Largest number is 24 (2nd, 3rd slots sum to 37): admissible tuples are (18, 19) and (17, 20)
• 25 (35): (17, 18), (16, 19) – (15, 20) is excluded since it repeats 15
• 26 (33): none – (16, 17) is excluded because it repeats 16
• 27 (31) and higher: none – one element in any tuple must be less than whatever has been defined as the smallest element, and so all tuples here are inadmissible.

Hence we see that there are only six lists of integers satisfying all the given conditions. $$12,20,21,21,22,22,22$$ $$13,19,20,21,22,22,23$$ $$14,18,19,21,22,22,24$$ $$14,17,20,21,22,22,24$$ $$15,17,18,21,22,22,25$$ $$15,16,19,21,22,22,25$$