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!

#Combinatorics #FibonacciNumbers #BinaryNumbers

Easy Math Editor

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Comments

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.

Pranjal Jain - 6 years, 9 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}`	$\boxed{123}$