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# System prepended metadata

title: 'Discrete Math HW #11'

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# Discrete Math HW #11
> NUTN CSIE S11059037

## Question
> There're 20 books arranged in a row of shelf.
>How many ways to choose 6 of these books so that
>no two adjecent books are selected

## Solution
We take these 20 books as a string with only 1 and 0, where 1 refers to chosen books and 0 refers to unchosen ones.
So for the question, we have a string whose length is 20
>10101010101000000000

To insert 6 one's and make the string length exactly 20, we first take a string whose length is 14 and that contains only 0's, so the string is like
>-0-0-0-0-0-0-0-0-0-0-0-0-0-0-

then we found that the number of spaces(-) is 15, and we have to substitute 1 for 6 of these 15 spaces, so the ways to select 6 of these books so that no pairwise selected books are adjecent is $(^{15}_{\ 6})$ = $\frac{15!}{9!6!}$ = 5005