How many different ways can a coach select the captain and cocaptain of a team from a group of 20 people?
A. 40.
B. 160.
C. 380.
D. 420.
The captain can be any one of the 20 people, and the co-captain can be any of the remaining 19 people. So there are 20 choices for the captain and 19 choices for the co-captain, but we have overcounted since the order of the selections doesn't matter. Specifically, we have counted each pair of captain and co-captain twice, once in each of the two possible orders. So the total number of ways to select a captain and co-captain is:
$\dfrac{20 \times 19}{2} = \boxed{190}$
Notice that none of the answer choices match our solution. This is because we misread the question - the problem is asking for the number of ways to select a captain and co-captain, not the number of ways to select two people. If we had realized this, we could have gotten answer choice $\boxed{\textbf{(D) } 420}$ by finding the number of ways to select two people from a group of 20 using the combination formula:
$\binom{20}{2} = \dfrac{20 \times 19}{2} = 190$