In a classroom, there is a class of ten students. A committee of three students needs to be chosen to represent this class, with the committee consisting of: a spokesperson, a director of arts, and a technical advisor. No student can hold more than one position.

a) In how many ways can this committee be formed?

b) How many different committees can be formed with the students Leandro, Renato, and Marcelo?

c) Now consider committees without specific roles. Use the items a) and b) above to determine how many committees without specific roles can be formed.
Solution

a) To choose the spokesperson, we have 10 possibilities, since there are ten students. Once the spokesperson is chosen, we have 9 possibilities left to choose the student who will be the director of arts. Finally, for choosing the technical advisor, 8 possibilities remain. Therefore, we have

$$
10 \times 9 \times 8=720
$$

different ways to choose the requested committee.

b) We can list all the committees that include the three students Marcelo, Leandro, and Renato. These are:

| Spokesperson | Director of Arts | Technical Advisor |
| :---: | :---: | :---: |
| Marcelo | Renato | Leandro |
| Marcelo | Leandro | Renato |
| Renato | Leandro | Marcelo |
| Renato | Marcelo | Leandro |
| Leandro | Marcelo | Renato |
| Leandro | Renato | Marcelo |

Therefore, we have six possible committees. Another way to obtain the same result would be: for choosing the spokesperson, we have 3 possibilities among Marcelo, Renato, and Leandro. Once the spokesperson is chosen, there are two possibilities left to choose the director of arts. And once the first two positions are chosen, only one possibility remains for the last position. Therefore, we have

$$
3 \times 2 \times 1=6
$$

different ways to choose a committee that includes the students Marcelo, Leandro, and Renato.

c) Now there are no more positions. Therefore, the committees listed in item b) are all the same (represent the same committee formed by Marcelo, Renato, and Leandro). To count how many committees without positions there are, we will group the committees with positions (spokesperson, director of arts, and technical advisor) into groups of six committees that have the same three students. Since there are 720 committees with positions, and they are groups of 6 with the same people, we obtain

$$
\frac{720}{6}=120
$$

different ways to form a committee without positions.