Question

Ten students audition for 3 different parts in a play in how many ways can the director fill these roles ?

Answer 1

The 3 roles can be filled in 720 different ways

Explanation

The director has to choose 3 students out of a group of 10 students. In each possible arrangement, which role is assigned to each selected student is also important. For example, is not the same if student A is the main role and student B is a supporting character that the inverse case. In consequence, the order of each group of students matters. Then, we need to use permutations to answer this question.

The number of permutations of n things chosen r at a time is calculated as follows:

`nPr=(n!)/((n-r)!)`

In this case, we need to select r = 3 students from a total of n = 10 students, that is,

`\begin{gathered} _(10)P_3=(10!)/((10-3)!) \\ _(10)P_3=(10!)/(7!) \\ _(10)P_3=720 \end{gathered}`