If a man climbs either one step or two steps at a time, the total number of ways in which he can climb up a staircase of 2n steps (starting from the bottom) is

Options

A. `sum_(r=0)^(n).^(2n-r)C_(r)`
B. `sum_(r=0)^(2n).^(2n+r)C_(r)`
C. `sum_(r=0)^(2n).^(2n-4)C_(r)`
D. `sum_(r=0)^(n).^(n-4)C_(r)`

Answer

To solve the problem of how many ways a man can climb a staircase of \(2n\) steps by taking either one step or two steps at a time, we can use a combinatorial approach.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The man can take either 1 step or 2 steps. If he takes \(x\) steps of 1 step each and \(y\) steps of 2 steps each, the total number of steps taken can be expressed as:
\[ x + 2y = 2n \]
Here, \(x\) is the number of single steps, and \(y\) is the number of double steps.

2. **Expressing the Total Steps**:
From the equation \(x + 2y = 2n\), we can express \(x\) in terms of \(y\):
\[ x = 2n - 2y \]
This means for every \(y\) (number of double steps), the corresponding \(x\) (number of single steps) can be calculated.

3. **Finding the Total Combinations**:
The total number of steps taken is \(x + y\). Thus, the total number of steps is:
\[ x + y = (2n - 2y) + y = 2n - y \]
The total number of ways to arrange these steps can be calculated using combinations. The number of ways to arrange \(x\) single steps and \(y\) double steps is given by:
\[ \frac{(x + y)!}{x! \cdot y!} = \frac{(2n - y)!}{(2n - 2y)! \cdot y!} \]

4. **Summing Over All Possible Values of \(y\)**:
Since \(y\) can take values from \(0\) to \(n\) (because if he takes \(n\) double steps, he cannot take any single steps), we sum over all possible values of \(y\):
\[ \text{Total Ways} = \sum_{y=0}^{n} \frac{(2n - y)!}{(2n - 2y)! \cdot y!} \]

5. **Using the Binomial Coefficient**:
The expression can be simplified using the binomial coefficient:
\[ \text{Total Ways} = \sum_{y=0}^{n} \binom{2n - y}{y} \]
This is equivalent to the \(n\)-th Fibonacci number \(F_{n+1}\) when \(n\) is the number of double steps.

6. **Final Result**:
Therefore, the total number of ways to climb a staircase of \(2n\) steps is given by:
\[ F_{n+1} \]