A man is at the origin on the x-axis and takes a unit step either to the left or the right. He stops after 5 steps or if he reaches 3 or -2. Number of ways in which he

Options

A. Reaches -2 is 3
B. reaches 3 is 4
C. stops exactly after taking 5 steps is 12
D. can perform the experiment is 20

Answer

To solve the problem, we need to analyze the man's movement along the x-axis, where he can take steps to the left or right, and determine the number of ways he can reach certain positions or stop after a specific number of steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The man starts at the origin (0) on the x-axis. He can take a step to the left (-1) or to the right (+1). He stops after 5 steps or if he reaches the positions 3 or -2.

2. **Finding Ways to Reach -2**:
- The man can reach -2 in two steps:
- Directly: 0 → -1 → -2 (1 way)
- The man can also reach -2 in 5 steps:
- Possible sequences include:
- 0 → 1 → 0 → -1 → 0 → -2 (1 way)
- 0 → -1 → 0 → -1 → -2 (1 way)
- Total ways to reach -2 = 1 (direct) + 2 (5 steps) = 3 ways.

3. **Finding Ways to Reach 3**:
- The man can reach 3 in 4 steps:
- Possible sequences include:
- 0 → 1 → 2 → 3 (1 way)
- 0 → 1 → 0 → 1 → 2 → 3 (1 way)
- 0 → -1 → 0 → 1 → 2 → 3 (1 way)
- Total ways to reach 3 = 4 ways.

4. **Finding Ways to Stop Exactly After 5 Steps**:
- The maximum position the man can reach after 5 steps is 5 (if he goes all right).
- The minimum position he can reach is -5 (if he goes all left).
- The only possible positions after 5 steps are from -5 to 5.
- The number of ways to reach any position after 5 steps can be calculated using combinations:
- For 5 steps, the number of ways to choose steps left (L) and right (R) can be expressed as:
- Total ways = \( \binom{5}{2} \) (choosing 2 left steps out of 5)
- This equals \( \frac{5!}{2!3!} = 10 \).
- However, since we need to consider all combinations of left and right, we multiply by 2 (for each step being either left or right):
- Total ways = \( 2 \times 10 = 20 \).

5. **Final Count of Ways**:
- The total number of ways to perform the experiment is 20.

### Summary of Results:
- Number of ways to reach -2: 3
- Number of ways to reach 3: 4
- Number of ways to stop exactly after 5 steps: 20

### Conclusion:
The correct options based on the analysis are:
- Reaches -2 in 3 ways (Correct)
- Reaches 3 in 4 ways (Correct)
- Stops exactly after taking 5 steps: 20 ways (Correct)