Question #ba20b

Mar 21, 2017

The number of combinations is $3003$

Explanation:

There are 5 men on a basketball team, and the coach has his choice of 15 men. There is no definition given of the position each man will be playing nor the relative importance of each player such as captain or forward.

This means we are working with a "combination" of $5$ players of equal status taken out of a total of $15$ players available.

Our possibilities start with $15$ available players then go down consecutively by $1$ as each player is chosen so we get $15 \cdot 14 \cdot 13 \cdot 12 \cdot 11$ possibilities by the time we arrive at the fifth team player.

But as we add players to the team, we also reduce the number of positions left, so as each player is added to the team our possibilities are further reduced: $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5$

Thus, for $15$ available players making a $5$ player team,

the combination looks like $C \left(15 , 5\right) = \frac{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

$C \left(15 , 5\right) = \frac{\cancel{15} \cdot \cancel{14} \cdot 13 \cdot \cancel{12} \cdot 11 \cdot 7 \cdot 3}{\cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot 1} = 3003$