Question

At the county fair, animals are judged for the quality of their breeding and health. The animal pens are arranged in an array, with one animal in each pen. A barn can hold at most 10 rows of pens and at most 6 pens in each row, with room for people to walk around them. What different ways can the planners of the county fair arrange the pens for the horses and cows in the same barn?

Answer 1

36045979200 + 18595558800 + 8835488640 +...

Explanation:

We suppose that we have 60 animals (between cows and horses) and we are going organize the animals in the pens in the indicated distribution.

First we selected six animals (of the set of 60 animals) for the first row through the combinatorial.

`60C6 = (60!)/((60-6)!6!) = 50063860`

Now such as the order of the animals in each row matter, we must use a permutation, thus:

6P6 = 6! = 720

Then for the first row we have 720*50063860 = 36045979200 possibilities

The proccess for locating the animals in the other rows is similar only the amount of animal is decreasing, thus.

Second row :

`54C6 = (54!)/((54-6)!6!) = 25827165`

25827165*720 = 18595558800

Third row:

`48C6 = (48!)/((48-6)!6!) = 12271512`

12271512*720 = 8835488640

And the process continues until the ten row.