How many different sets of 6 and 7 different numbers can we list out from 11,13,18,19,19,20,23,25? 
How many different sets of 6 and 7 different numbers can we list out from 11,13,18,19,19,20,23,25? 

Please no repeating in any case, if the numbers appeared in a set are exactly the same as another but different order is not counted. For example, 11,13,18,19,20 is the same as 13,18,11,20,19. I had hard time listing them all out cause everytime when I look back there were always some sets repeat. Both 6 and 7 number of sets please.
 A: 
Be aware that set means no two elements are the same.
That repeated 19 should make no difference.

Here are 6 number sets (total 7 of them):
{     13, 18, 19, 20, 23, 25 }
{ 11,     18, 19, 20, 23, 25 }
{ 11, 13,     19, 20, 23, 25 }
{ 11, 13, 18,     20, 23, 25 }
{ 11, 13, 18, 19,     23, 25 }
{ 11, 13, 18, 19, 20,     25 }
{ 11, 13, 18, 19, 20, 23     }

There is only this one 7 number set:
{ 11, 13, 18, 19, 20, 23, 25 }

Is there a mistake in the question?
Maybe repeated 19 should be some other number.

A: If a 'set' could have both 19s at the same time, then the easiest way to do this would be to start with the set of the six numbers 11, 13, 18, 20, 23, 25, adding that to the list, and then add six more with one of the numbers replaced with 19, and then it gets tricky, you need to find each combination of two 19's in the six positions, the best way to do that is just go through and check each individual one of these each time. that gives you 1 + 6 + 15 = 22 entries, and then you start with 11, 13, 18, 20, 23, 25, 19 for seven, add it to the list and then the six with every other number except the 19 replaced by 19, and that will give you 1 + 6 = 7 entries for seven, giving you a total of 22 + 7 entries, which should equal 39 entries total.
I haven't verified this solution as of yet, so please correct me if I am wrong.
