A hospital has the following employees:

Sara Dores da Costa: rheumatologist

Iná Lemos: pulmonologist

Ester Elisa: nurse

Ema Thomas: orthopedist

Ana Lisa: psychoanalyst

Inácio Filho: obstetrician

a) In how many ways can the employees form a line?

b) In how many ways can the same employees sit at a round table? Remember that, at a round table, if everyone moves to the chair on their left, the table remains the same!

c) And in how many ways can the employees form a committee consisting of a president, vice-president, and alternate?
Solution

a) To be the first in line, we can choose any one of the six employees. Therefore, there are 6 possibilities. Once the first in line is chosen, there are five employees left to be chosen as the second in line (since one has already been chosen). For the third position, we have 4 possibilities, and so on. Therefore, there are

$$
6 \times 5 \times 4 \times 3 \times 2 \times 1=720
$$

distinct ways to organize the line.

b) At a round table, only the relative position matters, as stated in the problem. If we move each employee to the chair on their left, the table remains the same. How many times can we do this? Six times, since there are six employees. If we number the chairs, we would have the same answer as in the previous part, 720. Since we can rotate the people to the adjacent chair 6 times, we conclude that the number of ways to seat the employees at the table is $720 / 6=120$.

c) We can choose the president in 6 ways, since there are 6 employees. For each of these choices, there are 5 possibilities to choose the vice president. And for each of these, 4 possibilities for the alternate. Therefore, there are $6 \times 5 \times 4=120$ ways to choose the committee.