In how many ways can 30 members of a club select a president, a vice-president and a secretary?

Answer 1

Assume any of the 30 members of the club have an equal chance of being selected.

Then the choice for President is 30 choose 1 = 30.

Now there are 29 members remaining for VP, or 29 choose 1 = 29.

Finally there are 28 members from which to select the Secretary, or 28 choose 1 = 28.

The number of ways to select all three is 30*29*28 = 24,360

Answer 2

Initially, there are 30 people to pick as president. After you pick the president, there are now 29 people that can be picked as VP. After you pick the VP, there are now 28 people that you can pick for secretary. That give 30*29*28 or 24,360 possible choices.

This is a permutation because order makes a difference (meaning, if Alice, Fred, and Jennifer are club members then having Fred as president, Alice as VP, and Jennifer as secretary is different than having Alice as president, Jennifer as VP and Fred as secretary). Each order selected is different set of officers.

In contrast, if you were selecting a subcommittee in the club and Alice, Fred, and Jennifer were selected to be in the subcommittee, then it would not make a difference what order they were selected because selecting Alice, Fred, and Jennifer results in the same sub-committee as selecting Jennifer, Alice, and Fred. When order does not matter, it is called a combination and there are fewer possibilities associated with that.

The equation for a permutation (the case in this problem) uses two things. The number of in the group (n) and the number being selected (r). In this case n = 30 (30 members in the club) and r = 3 (3 officers are being selected. The equation is:

P(n, r) = n!/(n - r)! where "!" means factorial

In this case P(30, 3) = 30!/(30 - 3)! = 30!/27! = (30*29*28*27*26 . . . *2*1)/(27*26*25*24 . . .*2*1) = 30*29*28

If you have a TI-84 calculator, you can use the ₃₀P₃ function and it will result is the 24,360 answer. If you need help with that, let me know.