1:A train is to consist of three flatcars, five tankers, six boxcars, an engine and a caboose.In how many ways can the train be made ?
To find the number of ways the train can be made, we can use the concept of permutations.
First, let's choose the position for the engine. Since there are 11 positions in total (3 flatcars + 5 tankers + 6 boxcars + 1 engine + 1 caboose), we have 11 choices for the engine.
Next, we move on to the caboose. We have 10 remaining positions to choose from, so there are 10 choices for the caboose.
Now, let's consider the flatcars. We have 3 flatcars, and we need to choose their positions from the remaining 9 positions (11 total positions minus the engine and the caboose). Since the order of the flatcars matters (if we place them differently, it will result in a different train configuration), we can use a permutation to calculate this. The number of ways to arrange the 3 flatcars is 9P3, which can be calculated as:
9P3 = 9! / (9 - 3)! = 9! / 6! = 9 * 8 * 7 = 504
For the tankers, we have 5 tankers, and we need to choose their positions from the remaining 6 positions. Again, since the order matters, we can use a permutation to calculate this. The number of ways to arrange the 5 tankers is 6P5, which can be calculated as:
6P5 = 6! / (6 - 5)! = 6! / 1! = 6
Lastly, for the boxcars, we have 6 boxcars, and we need to choose their positions from the remaining 1 position. Again, the order matters, so we have just 1 choice for the boxcars.
To calculate the total number of ways the train can be made, we multiply the number of choices for each component:
Total number of ways = 11 choices for the engine * 10 choices for the caboose * 504 choices for the flatcars * 6 choices for the tankers * 1 choice for the boxcars
Total number of ways = 11 * 10 * 504 * 6 * 1 = 3,364,640
Therefore, the train can be made in 3,364,640 different ways.
I will assume engine up front and caboose at the rear so the different ways relate to the other cars.
Now we do not know if the tankers are all picked up together and the flatcars together etc. or not. In practice that would be likely because they would come from different places, so I will assume that you have a group of 3, a group of 5 and a group of 6
Now those three groups can be arranged in 3! = 6 ways
If it does not matter what order the cars are in within the groups, you are finished. However if it does, we need to go on.
The permutations of 3 is 3! = 6
of 5 is 5! = 120
of 6 is 6! = 720
so we have the original 6 for the three groups times all the permutations within the groups
6*6*120*720