Question

Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:

At any stage, Ankita can move either one or two stairs up.
At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{\text{th}}$ stair. For example, $F(1)=1$, $F(2)=2$, $F(3)=3$. The value of $F(5)$ is ____________.

• A. 8
• B. 7
• C. 6
• D. 5

Hint:

Climbing problems with allowed steps $\{1,2\}$ follow the Fibonacci-type recurrence $F(N)=F(N-1)+F(N-2)$ with $F(1)=1$, $F(2)=2$. Compute iteratively to avoid mistakes.

Answer 1

The Correct Option is A

Step 1: Establish the recurrence.
To reach stair $N$ (with steps of size 1 or 2 only), Ankita’s last move must be: \begin{itemize}
from stair $N-1$ with a single step, or
from stair $N-2$ with a double step. \end{itemize} Therefore, the number of ways satisfies \[ F(N)=F(N-1)+F(N-2),\qquad N\ge 3, \] with the base cases given in the question: $F(1)=1$, $F(2)=2$.
Step 2: Compute successively up to $N=5$.
\[ \begin{aligned} F(3)&=F(2)+F(1)=2+1=3,
F(4)&=F(3)+F(2)=3+2=5,
F(5)&=F(4)+F(3)=5+3=8. \end{aligned} \]
Step 3: (Optional) Verify by enumeration.
Sequences of steps (1 = single step, 2 = double step) summing to 5: $11111$, $2111$, $1211$, $1121$, $1112$, $221$, $212$, $122$ $\Rightarrow$ $8$ ways. Final Answer: \[ \boxed{8} \]