Question

Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits $0,1$ or both such that no consecutive digits in them are $0$. Let $b_n=$ the number of such $n$-digit integers ending with digit $1$ and $c_n=$ the number of such $n$-digit integers ending with digit $0$. The value of $b_6$ is

• A. $7$
• B. $8$
• C. $9$
• D. $11$

Answer 1

The correct option is B $8$

To find $b_6$, we have to find all 6 digit numbers ending with '1' such that no consecutive digits are '0'. Some of the examples possible are :
1. 1 0 1 1 1 1
2. 1 0 1 0 1 1
3. 1 1 1 1 1 1
Three case possible:
1. One zero- It can be placed in any of the four places.
So, we get '4' such six digit numbers.
2. Two zeros- We get '3' such six digit numbers possible.
3. No zeros- We get only '1' such six digit number.
Hence $b_6=4+3+1=8$