A common mistake made on this problem is the following:
Choose 4 of the first 6 questions.
2) Then, you have 3 choices to make of the remaining 6
It seems as if we can do this in (
) = 300 ways.
But, as we can see, this answer is incorrect, by quite a bit.
We are actually counting some of the combinations of
questions more than once, that is why this answer is incorrect.
Consider the combination 1, 2, 3, 4, 5, 7, and 8. We can arrive
at this combination in a couple ways:
1) Pick 1,2,3,4 of the first 6, then 5,7,8 of the remaining Qs.
2) Pick 1,2,3,5 of the first 6, then 4,7,8 of the remaining Qs.
We count these two possibilities separately in the count above.
But, we shouldn’t. Thus, we are over counting and end up with
an incorrect answer.
Combinations with Repetition