Question

In a train yard there are 5 tank cars, 11 boxcars, and 6 flatcars. How many ways can a train be made up consisting of 2 tank cars, 5 boxcars, and 3 flat cars? (Order is not important.)

Answer 1

There are 92,400 different ways a train can be made up using 2 tank cars, 5 boxcars, and 3 flatcars.

Step-by-step explanation:

To find the number of ways a train can be made up, we need to use combinations. We have 5 tank cars, 11 boxcars, and 6 flatcars, and we need to choose 2 tank cars, 5 boxcars, and 3 flatcars. The number of ways to choose the tank cars is given by C(5,2), the number of ways to choose the boxcars is C(11,5), and the number of ways to choose the flatcars is C(6,3). To find the total number of ways, we need to multiply these three combinations together:

C(5,2) * C(11,5) * C(6,3)

Using the combination formula, C(n,r) = n! / ((n-r)! * r!), we can calculate the values:

C(5,2) = 5! / ((5-2)! * 2!) = 5! / (3! * 2!) = 10

C(11,5) = 11! / ((11-5)! * 5!) = 11! / (6! * 5!) = 462

C(6,3) = 6! / ((6-3)! * 3!) = 6! / (3! * 3!) = 20

Now, we can multiply these values together:

10 * 462 * 20 = 92,400

Therefore, there are 92,400 different ways a train can be made up using 2 tank cars, 5 boxcars, and 3 flatcars.