Question

There are 4 copies of 5 different books in how many ways can they be arranged on a shelf

Answer 1

   

Let each different book be indicated as A, B, C, D and E.

Each book has 4 copies. So, one such arrangement is given below.

AAAABBBBCCCCDDDDEEEE

Here, we got a 20-letter word of 5 letters recurring 4 times each.

Now I'm changing the question as the following:

"How many meaningful and non-meaningful words can be formed by rearranging the letters in the word AAAABBBBCCCCDDDDEEEE?"

Here, the answer is,

$\ \ \ \ \ \boxed{\frac{20!}{4! \times 4! \times 4! \times 4! \times 4!}} \\ \\ \\ \Rightarrow\ \boxed{\frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 4 \times 3 \times 2 \times 4 \times 3 \times 2 \times 4 \times 3 \times 2}}$

$\Rightarrow\ \boxed{5 \times 19 \times 18 \times 17 \times 15 \times 14 \times 13 \times 11 \times 10 \times 7 \times 5} \\ \\ \\ \Rightarrow\ \boxed{\bold{3,05,54,02,35,000}}$

Thus we can arrange 4 copies of 5 different books on a shelf in 3,05,54,02,35,000 ways.

Hope this helps you.

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The answer is on my own words and not from any source.

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