Question

If $N$ is the number of ways in which a person can walk up a stairway which has $7$ steps if he can take $1$ or $2$ steps up the stairs at a time, then the value of $N/3$ is

Answer 1

$x$ denotes the number of times he takes unit step and $y$ denotes the number of times he takes $2$ steps, then $x+2y=7$,
Then we must have $x=1,3,5,7$
If $x=1$, the steps will be $1,2,2,2$
$\Rightarrow$ number of ways $=\frac{4!}{3!}=4$

If $x=3$ the steps will be $1,1,1,2,2$
$\Rightarrow$ number of ways $=\frac{5!}{2!\cdot 3!}=10$

If $x=5$ the steps will be $1,1,1,1,1,2$
$\Rightarrow$ number of ways ${}^6{C}_1=6$

If $x=7$ the steps will be $1,1,1,1,1,1,1,1$
$\Rightarrow$ number of ways ${}^7{C}_0=1$
Hence total number of ways $=N=21$
$\Rightarrow N/3=7$