John has 4 math books, 6 biology books,, and 8 statistics books to arrange on a shelf. In how many ways can he arrange the books so that books on the same subject are grouped together?
Use either permutations or combinations
4!*6!*8!*3!
To solve this problem, we will consider arranging each group of books separately, and then multiply the results together.
First, let's consider the math books. John has 4 math books, which can be arranged in 4! (4 factorial) ways. Factorial means multiplying the number by all positive integers smaller than it. So, 4! = 4 x 3 x 2 x 1 = 24.
Next, let's consider the biology books. John has 6 biology books, which can be arranged in 6! ways, or 6 x 5 x 4 x 3 x 2 x 1 = 720.
Finally, let's consider the statistics books. John has 8 statistics books, which can be arranged in 8! ways, or 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320.
Now, since each group of books can be arranged independently of the others, we need to multiply the number of arrangements for each subject together. So, the total number of arrangements is:
24 x 720 x 40,320 = 55,147,520.
Therefore, there are 55,147,520 ways for John to arrange the books on the shelf so that the books on the same subject are grouped together.