Answer:
C
Explanation:
first of all the situation is simply :
for the first position you have the full choice of the 30 members.
but then for the other positions you have taken 1 member out of the pool, and we have only 29 left to pick from.
after picking the second position, you have taken 2 members out of the pool, and we have only 28 choices for the 3rd position.
that is what the answer is
30×29×28
now to the theory :
well, the sequence is important.
if you go for the combinations of picking 3 people out of 30, you don't care, if m1 is in p1, m2 in p2, and m3 in p3, or m1 in p2, m2 in p3 and m3 in p1. since it is the same group of m1, m2, m3, both cases would be the same for combinations.
but for our purpose here, they are not the same.
it is important who is president, vice president and treasurer.
like in the Olympic games it makes a big difference, who wins gold, silver and bronze. it is not just to win a medal, and nobody cares about the color ...
they do, and therefore the sequence matters.
and therefore it is a permutation.
basically, the last division by 3×2×1 in the combinations is simply the removal of all the different sequences inside a group of 3.
as, in how many ways can you order 3 items ? 6 : 3×2×1.
3 options for the first, then 2 options for the second, and then one remaining for the third.
but again, if the sequence inside the group of 3 is important, you need to keep these extra 6 options per group of 3 in the calculation.
if the description would have been :
the club votes 3 members as board, and all 3 board members can fulfill all 3 roles at any time as needed, then the sequence would not matter, and it would be combinations.
but the sequence matters, and it is permutations.