Question

1. freight train cars in a train yard there are 4 tank cars, 12 boxcars, and 7 flatcars. how many ways can a train be made up consisting of 2 tank cars, 5 boxcars, and 3 flatcars? (in this case, order is not important.)

Explanation:

Step1: Calculate the number of ways to choose tank cars

The number of ways to choose \(2\) tank cars out of \(4\) is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 4\) and \(r=2\).

$$C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=\frac{4\times3}{2\times 1}=6$$

Step2: Calculate the number of ways to choose boxcars

The number of ways to choose \(5\) boxcars out of \(12\) is \(n = 12\), \(r = 5\)

$$C(12,5)=\frac{12!}{5!(12 - 5)!}=\frac{12\times11\times10\times9\times8\times7!}{5!\times7!}=\frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}=792$$

Step3: Calculate the number of ways to choose flatcars

The number of ways to choose \(3\) flatcars out of \(7\) is \(n=7\), \(r = 3\)

$$C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5\times4!}{3!\times4!}=\frac{7\times6\times5}{3\times2\times1}=35$$

Step4: Use the multiplication principle

Since the choices of tank - cars, box - cars, and flat - cars are independent events, the total number of ways to make up the train is the product of the number of ways of each choice.

$$N = C(4,2)\times C(12,5)\times C(7,3)=6\times792\times35$$

$$N=6\times27720=166320$$

Answer:

\(166320\)