# If last year's champs, the Anows, must be in the first division and the current league leaders, the Bows, must be in the second division, in how many ways can the selection be made?

## There are 14 teams competing in the Bilton Community Darts League. Teams are to be selected for the first and second divisions in the upcoming tournament. Each division is to have seven teams.

Jun 16, 2018

The selection can be made in $924$ ways

#### Explanation:

In the first division there must be the Arrows plus 6 other teams from the total of 14 excluding the Bows (who must be in the second division) and, of course, the Arrows (who can not be counted in the division more than once).

That is the first division is composed of the Arrows plus $6$ out of $12$ other teams.

There are color(white)(0)_12C_6=(12!)/(6! * (12-6)!)

$\textcolor{w h i t e}{\text{XXXXXXXXX}} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$

(after some cancelling and multiplying on scrap of paper)
$\textcolor{w h i t e}{\text{XXXXXXXXX}} = 924$ ways to do this.

Once the first division has been established there is only $1$ way to fill the second division.