Four men A,B,C and D, line up in a row. What is the probability that man A is at either end of the row?

Options

A. `1/2`
B. `1/3`
C. `1/4`
D. `1/6`

Explanation

To solve the problem of finding the probability that man A is at either end of the row when four men A, B, C, and D line up, we can follow these steps:

### Step 1: Calculate the total number of arrangements of the four men.
The total number of arrangements of four men can be calculated using the factorial of the number of men. Since there are 4 men, the total arrangements can be calculated as:

\[ \text{Total arrangements} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 2: Determine the favorable arrangements where A is at either end.
Man A can occupy either the first position or the last position in the row. We will calculate the arrangements for both scenarios:

1. **A at the first position**: If A is at the first position, we have 3 remaining men (B, C, D) who can occupy the other 3 positions. The number of arrangements for B, C, and D is:

\[ 3! = 3 \times 2 \times 1 = 6 \]

2. **A at the last position**: Similarly, if A is at the last position, the arrangements for B, C, and D remain the same:

\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 3: Calculate the total favorable arrangements.
Since A can be at either end, we add the two scenarios together:

\[ \text{Total favorable arrangements} = 6 + 6 = 12 \]

### Step 4: Calculate the probability.
The probability that man A is at either end of the row can be calculated by dividing the number of favorable arrangements by the total arrangements:

\[ \text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{12}{24} = \frac{1}{2} \]

### Final Answer:
The probability that man A is at either end of the row is \(\frac{1}{2}\).

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