## If Jacob owns 12 books, how many different ways can he arrange 3 of these books on a single bookshelf

Question

If Jacob owns 12 books, how many different ways can he arrange 3 of these books on a single bookshelf

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3 months 2022-02-06T15:23:29+00:00 1 Answer 0 views 0

We will think as follows. Recall that at first, he chooses 3 books. Then, we want to see in how many ways he can arrange this books. For the first book, he has 3 options, for the second one 2 and for the third 1. Then, the total number of ways in which he can arrange 3 books is 3!. So we know that after choosing 3 books, he can arrange them in 3! ways. So, we want to check the total number of ways he can choose 3 books out of 12. That is given by the binomial coefficient $$\binom{12}{3}$$. Then, the total number of ways of arranging 3 books is
$$\binom{12}{3}\cdot 3! = \frac{12!}{(12-3)! 3!}\cdot 3! = \frac{12!}{(12-3)!} = 1320$$