Question

A baseball manager has 10 players of the same ability. How many 9-player starting lineups can he create?
a) 10
b) 90
c) 100
d) 120

Answer 1

Answer:

$10$.

Step-by-step explanation:

The question is asking for the number of possible ways to choose $9$ players out of $10$. Since the question implies that the ordering of the chosen players does not matter, this number of possible lineups can be found using the combination formula.

The number of possible ways to choose $r$ items out of a total of $n$ options is:

$\displaystyle \frac{n!}{r!\, (n - r)!}$.

In this question, the goal is to choose $r = 9$ players out of a total $n = 10$. Hence, the possible number of lineups would be:

$\begin{aligned} \frac{n!}{r!\, (n - r)!} &= \frac{10!}{9!\, (10 - 9)!} \\ &= \frac{10 \times 9 \times 8 \times \left.\cdots\right. \times 1}{(9\times 8 \times \left.\cdots\right. \times 1) \, (1)} \\ &= 10 \end{aligned}$.

In other words, there are a total of $10$ possible lineups.