Question

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 is

Answer 1

$x$ denotes the number of times he can take unit step and $y$ denotes the number of times he can take $2$ steps, then $x+2y=7$,
Then $x$ can be $x=1,3,5,7$
If $x=1$, the steps will be $1222$
$\Rightarrow$ number of ways $=\frac{4!}{3!}=4$
If $x=3$, the steps will be $11122$
$\Rightarrow$ number of ways $=\frac{5!}{2!.3!}=10$
If $x=5$, the steps will be $111112$
$\Rightarrow$ number of ways $=\frac{6!}{5!}=6$
If $x=7$, the steps will be $1111111$
$\Rightarrow$ number of ways $=\frac{7!}{7!}=1$
Hence total number of ways $=4+12+6+1=21$