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Playing with Numbers

I was recently doing a combinatorics problem, where we had to select 4 books from a set of 10 ,where the selected books are not adjacent to each other. The author put forward the idea of using a 10-digit binary number and the books to be selected were 1's and the others were 0's. While playing with the idea I eventually found that(Image above)

and so on, and thus followed the Fibonacci Sequence! Does anyone know why it is? Please share you opinions!

Note by Siddharth G
3 years, 8 months ago

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Let $$a_{n}$$ be the number of n-digit binary numbers such that no 1's are together.

Now such number might end with 0 or 1

Case-1, It ends with 0 Now the number of such n-digit numbers will be $$a_{n-1}$$

Case-2, It ends with 1 Here, it cannot end with $$\boxed{11}$$ block. It must be ending with $$\boxed{01}$$ Number of such binary numbers will be $$a_{n-2}$$

Therefore $$a_{n}$$=$$a_{n-1}$$ + $$a_{n-2}$$ which is the condition for fibonacci series.

- 3 years, 5 months ago