Today, we will discuss about the Daily Magic Spell proposed on January 6, 2017.
Problem: Suppose you have a club of $20$ members. The club chooses four officers: President, Vice-President, Treasurer, and Secretary. They also choose someone to be in charge of fundraising. The four officers must all be different, but the member in charge of fundraising can be one of the officers. How many ways can we choose the four officers in the club with no restrictions?
Solution: We can first choose the President in the club. There are $20$ options.
After the President is chosen, when choosing the Vice-President, there are $19$ options.
We repeat this process until all of the positions are filled.
Therefore, there are $20 \times 19 \times 18 \times 17 = 116280$ ways to fill the four positions.
Additionally, we wish to appoint someone in the club to take charge of fundraising. This can be any one of the $20$ members in the club. Therefore, we have: $116280 \times 20 = 2325600$ ways to appoint four officers and a person in charge of fundraising.
The common approach that most students who answered this question incorrectly is most likely interpreting the problem incorrectly. Some users may not account for the possibility of one of four officials to be the member in charge of fundraising. Because of this, a common answer might be $20 \times 19 \times 18 \times 17 \times 16 = 1860480$.
Otherwise, this problem is a straight forward Product Rule problem (Counting Principle). Most students who have attempted this problem recognized this!
If you have any other approaches that you believe is worth discussing about, please let us know by posting here! We would greatly appreciate listening to your ideas!