The number of different sets created by 6 integers from a set of 10 is calculated as follows.
Choice of first integer is all 10 integers: 10
Now there are 9 integers left, so there are 10*9=90 sets of 2 integers.
Moving on to the third integer, we have 8 and so on.
Continuing until we have 6 integers: 10*9*8*7*6*5.
But in a set the order isn't important, just the combination so we need to divide by the number of arrangements of 6 integers = 6*5*4*3*2*1=720. Therefore the number of sets is 10*9*...*5/(720)=210.
There are only six integers between 10 and 15 so only one set of 6 integers is possible.