Question

a colleges baseball roster consists of 11 pitchers, 3 catchers, 8 infielders, and 5 outfielders. if the team manager randomly selects 9 players, what is the probability that he selects 1 pitcher, 1 catcher, 3 infielders, and 4 outfielders?
the probability is (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate total number of players

The total number of players on the roster is $11 + 3+8 + 5=27$.

Step2: Calculate number of ways to choose 9 - player team

The number of ways to choose 9 players out of 27 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 27$ and $r = 9$. So, $C(27,9)=\frac{27!}{9!(27 - 9)!}=\frac{27!}{9!18!}=\frac{27\times26\times\cdots\times19}{9\times8\times\cdots\times1}=4686825$.

Step3: Calculate number of ways to choose the specific - type players

The number of ways to choose 1 pitcher out of 11 is $C(11,1)=\frac{11!}{1!(11 - 1)!}=11$.
The number of ways to choose 1 catcher out of 3 is $C(3,1)=\frac{3!}{1!(3 - 1)!}=3$.
The number of ways to choose 3 infielders out of 8 is $C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56$.
The number of ways to choose 4 outfielders out of 5 is $C(5,4)=\frac{5!}{4!(5 - 4)!}=5$.
The number of ways to choose 1 pitcher, 1 catcher, 3 infielders, and 4 outfielders is the product of these combinations: $11\times3\times56\times5 = 9240$.

Step4: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So, $P=\frac{9240}{4686825}=\frac{616}{312455}$.

Answer:

$\frac{616}{312455}$