TITLE: How many subsets of four numbers from the set are there in which the sum of the largest and smallest number in the subset is 15?
QUESTION [0 upvotes]: How many subsets of four numbers from the set $2,3,4,5,6,7,8,9,10,11,12,13,14$ are there in which the sum of the largest and smallest number in the subset is $15$?
The answer is: $C(10,2)+C(8,2)+C(6,2)+C(4,2)+C(2,2)$
I have no idea where these numbers are coming from? Can someone please help me to understand what the solution is doing. The choose $2$ is especially confusing because it says subsets of $4$ numbers.

REPLY [1 votes]: Let the the set is S={a,b,c,d} and it is given that $a+d=15$.
Now fisrt choose the smallest number in the given set of 13 integers.It
is 2 and according to 2 the largest number is $15-2=13$. Now fix 2 and 13 in the
place of a and d.Now we have S={2,b,c,13} and now choose b,c from the residue
set (besides 14){3,4,5,6,7,8,9,10,11,12}. The total number of ways doing this $\implies\;{10\choose2}$.The 
strategy is to choose the minimum and accordingly maximum will be chosen.Then 
Count the number of ways from the residue set={minimum,...,maximum}.Then add 
them for each minimum or maximum case, then you will have the answer of your
required question.
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