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# Combination

How many ways you can pick 5 books from 12 books such that no two are consecutive?

Note by Christian Lim
4 years, 9 months ago

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Let the books be numbered $$1$$ to $$12$$. Now after the selection process, we label each book as $$A$$ or $$B$$ according to the following rule:- if a particular book is chosen, it is labelled $$A$$ and if it is not chosen, it is labelled $$B$$. Then we write the $$A$$ $$B$$ sequence. Note that each sequence corresponds to an unique selection of books. For example, the sequence $$ABABABABBBBB$$ means that book $$1$$ is chosen, book $$2$$ isn't, book $$3$$ is chosen, book $$4$$ isn't, book $$5$$ is chosen, book $$6$$ isn't, book $$7$$ is chosen, and books $$8$$ to $$12$$ aren't. Then our total number of acceptable permutations will be the number of ways of permuting $$5$$ $$A$$s and $$7$$ Bs such that no two $$A$$s are beside one another. To do this, place the 7 $$B$$s in gaps, like this $$_B_B_B_...$$. Now there are $$8$$ possible gaps and $$5$$ gaps have to be filled by $$A$$s. This can be done in $${8 \choose 5}$$ ways.

- 4 years, 9 months ago

I got the answer as 56, i.e. 8C5.

- 4 years, 9 months ago

can you explain the que, i can't understand what do u mean by "no 2 are consecutive"

Well.. the books are stacked side by side.. you're suppose to choose 5 that are not next to each other. I would solve this using complementary counting and then applying the principle of inclusion and exclusion. (I haven't tried it out yet.. so I'm not sure if it'll work)

- 4 years, 9 months ago

can you explain the que, i can't understand what do u mean by "no 2 are consecutive"

- 4 years, 9 months ago