# Question f07f8

Aug 3, 2017

(30!)/(6!(30-6)! ) = 593,775

#### Explanation:

20% of the students will be evaluated. The order in which the teacher sees them is not important, as long as $6$ of them are chosen.

There are $30$ choices for the first student, then $29$ for the second, then $28$ and so on until $6$ students are chosen.

The total number of possible groups of $6$ is

$30 \times 29 \times 28 \times 27 \times 26 \times 25$

This can also be written as (30!)/((30-6)!)

$\frac{30 \cdot 29 \cdot 28 \cdot 27 \cdot 26 \cdot 25 \cancel{\cdot 24 \cdot 23 \cdot 22 \ldots} .}{\cancel{24 \cdot 23 \cdot 22 \cdot \ldots}}$

However within this number of groups, the same groups just in a different order are included, so we need to reduce the number by the number of ways of arranging $6$ students which is 6!

The number of different groups of $6$ students is:

(30!)/(6!(30-6)! ) = 593,775#