Combination

Please Solve:

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

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

Sreejato Bhattacharya - 7 years, 12 months ago

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

Vikram Waradpande - 8 years ago

My answer is 41..please comment about my answer :))

Christian Lim - 8 years ago

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

Aditya Jain - 8 years 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)

Taehyung Kim - 7 years, 12 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$