a baseball team consists of three outfielders, four infielders, a pitcher, and a catcher. assuming that the…

a baseball team consists of three outfielders, four infielders, a pitcher, and a catcher. assuming that the outfielders are indistinguishable from each other and the infielders are indistinguishable from each other, how many batting orders are possible?\nthere are □ possible batting orders.\n(type a whole number.)

a baseball team consists of three outfielders, four infielders, a pitcher, and a catcher. assuming that the outfielders are indistinguishable from each other and the infielders are indistinguishable from each other, how many batting orders are possible?\nthere are □ possible batting orders.\n(type a whole number.)

Answer

Explanation:

Step1: Calculate the total number of players

The total number of players is (3 + 4+1 + 1=9) players.

Step2: Use the formula for permutations of multi - sets

The formula for the number of permutations of a multi - set with (n) objects, where there are (n_1) of one kind, (n_2) of another kind,(\cdots), (n_k) of the (k)th kind is (\frac{n!}{n_1!n_2!\cdots n_k!}). Here (n = 9) (total number of players), (n_1=3) (number of out - fielders) and (n_2 = 4) (number of in - fielders). The number of batting orders is (\frac{9!}{3!4!}) We know that (n!=n\times(n - 1)\times\cdots\times1), so (9! = 9\times8\times7\times6\times5\times4\times3\times2\times1=362880), (3! = 3\times2\times1 = 6) and (4! = 4\times3\times2\times1=24) [ \begin{align*} \frac{9!}{3!4!}&=\frac{9\times8\times7\times6\times5\times4!}{3\times2\times1\times4!}\ &=\frac{9\times8\times7\times6\times5}{6}\ &=9\times8\times7\times5\ &=2520 \end{align*} ]

Answer:

(2520)