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 $8$

For $b_n$
First and last place are fixed by $1$
So case $\left(1\right)$ if only one zero is used such cases $={}^{n-2}{C}_1$
case $\left(2\right)$ if two zeros are used then the position of two zeros such that no two zeros are cosecutive $={}^{n-3}{C}_2$
case $\left(3\right)$ if three zeros are used then the position of three zeros such that no two zeros are cosecutive $={}^{n-4}{C}_3$
So, $b_n=1{+}^{n-2}{C}_1{+}^{n-3}{C}_2{+}^{n-4}{C}_3{+}^{n-5}{C}_4+\dots$
For $b_6{=}^4{C}_1{+}^3{C}_2+1=8$