A girl has to climb 12 steps. She climbs in either a single step or 2 steps simultaneously. In how many ways can she do it?

Options

A. 233
B. 233
C. 322
D. 232

Answer

To solve the problem of how many ways a girl can climb 12 steps by taking either 1 step or 2 steps at a time, we can use a combinatorial approach.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The girl can take either 1 step or 2 steps. We need to find out how many distinct ways she can reach the 12th step.

2. **Defining Variables**:
Let \( f(n) \) represent the number of ways to climb \( n \) steps. The girl can reach the \( n \)-th step by either:
- Taking a single step from the \( (n-1) \)-th step, or
- Taking two steps from the \( (n-2) \)-th step.

This gives us the recurrence relation:
\[ f(n) = f(n-1) + f(n-2) \]

3. **Base Cases**:
- If there are 0 steps, there is 1 way to stay at the ground (do nothing): \( f(0) = 1 \)
- If there is 1 step, there is also 1 way to climb it: \( f(1) = 1 \)

4. **Calculating Further Values**:
We can calculate \( f(n) \) for \( n = 2 \) to \( n = 12 \) using our recurrence relation:
- \( f(2) = f(1) + f(0) = 1 + 1 = 2 \)
- \( f(3) = f(2) + f(1) = 2 + 1 = 3 \)
- \( f(4) = f(3) + f(2) = 3 + 2 = 5 \)
- \( f(5) = f(4) + f(3) = 5 + 3 = 8 \)
- \( f(6) = f(5) + f(4) = 8 + 5 = 13 \)
- \( f(7) = f(6) + f(5) = 13 + 8 = 21 \)
- \( f(8) = f(7) + f(6) = 21 + 13 = 34 \)
- \( f(9) = f(8) + f(7) = 34 + 21 = 55 \)
- \( f(10) = f(9) + f(8) = 55 + 34 = 89 \)
- \( f(11) = f(10) + f(9) = 89 + 55 = 144 \)
- \( f(12) = f(11) + f(10) = 144 + 89 = 233 \)

5. **Final Result**:
The total number of ways for the girl to climb 12 steps is \( f(12) = 233 \).

### Conclusion:
Thus, the girl can climb the 12 steps in **233 different ways**.

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