Anita and Binita stand in a line with 7 other people. What is the probability that there are 4 persons between them?

Answer

To solve the problem of finding the probability that Anita and Binita stand in a line with 7 other people such that there are exactly 4 persons between them, we can follow these steps:

### Step 1: Understand the total number of people
Anita and Binita are two people, and there are 7 other people. Therefore, the total number of people is:
\[ \text{Total people} = 2 + 7 = 9 \]

### Step 2: Identify the arrangement conditions
We want to find the arrangements where there are exactly 4 people between Anita and Binita. This means that if we denote the positions of Anita and Binita as A and B, the arrangement must look like:
\[ A \_ \_ \_ \_ B \]
or
\[ B \_ \_ \_ \_ A \]
where the underscores represent the 4 people in between.

### Step 3: Determine possible positions for A and B
The positions of A and B can be represented in a line of 9 positions. The possible positions for A and B while maintaining 4 people in between them are:
1. A at position 1 and B at position 6
2. A at position 2 and B at position 7
3. A at position 3 and B at position 8
4. A at position 4 and B at position 9

This gives us a total of 4 valid arrangements for A and B.

### Step 4: Arrange the remaining people
Once A and B are placed, we have 7 remaining positions (the 4 people in between and the 3 remaining positions) to fill with the other 7 people. The number of ways to arrange these 7 people in the remaining 7 positions is:
\[ 7! \]

### Step 5: Account for the arrangement of A and B
Since A and B can be arranged in two different ways (A can be before B or B can be before A), we multiply the arrangements by 2:
\[ \text{Total favorable arrangements} = 4 \times 7! \times 2 \]

### Step 6: Calculate the total arrangements of all people
The total number of arrangements of all 9 people (without any restrictions) is:
\[ 9! \]

### Step 7: Calculate the probability
The probability that there are exactly 4 persons between Anita and Binita is given by the ratio of favorable arrangements to total arrangements:
\[ P(\text{4 persons between A and B}) = \frac{4 \times 7! \times 2}{9!} \]

### Step 8: Simplify the probability
We know that:
\[ 9! = 9 \times 8 \times 7! \]
Thus:
\[ P(\text{4 persons between A and B}) = \frac{4 \times 7! \times 2}{9 \times 8 \times 7!} = \frac{8}{72} = \frac{1}{9} \]

### Final Answer
The probability that there are 4 persons between Anita and Binita is:
\[ \frac{1}{9} \]

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