Show that the number of ways n number of books can be arranged on a shelf such that two particular books are never together is (n-2)(n-1)!
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Answer:
(n-2)(n-1)!
Step-by-step explanation:
What is the number of ways in which n books can be arranged on a shelf so that two particular books shall not be together (the answer given in my book is (n-1) n, but I'm not able to understand it)?
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3 Answers
K S Narayanan, Practicing Cost Accountant
Answered June 11, 2018 · Upvoted by Luís Sequeira, PhD Mathematics, University of Lisbon (2001) and Terry Moore, M.Sc. Mathematics, University of Southampton (1968)
“n” distinct books can be arranged in a bookshelf in n! ways.
Let us assume the particular books are always together and is treated as 1 object.
So there are (n - 1) objects that can be arranged in (n - 1)! ways and the two books can be arranged between themselves in 2 ways.
=> Number of ways in which these two books will ALWAYS be together is :
2 (n - 1)!
Therefore, number of ways in which these two books will NEVER be together is :
n! - 2 (n - 1)!
= n(n - 1)! - 2 (n - 1)!
= (n - 2)(n - 1)!