There are three families in which 2 families has 3 member each and thrid family has 4 members. they are arranged in aline , then probability that members of same family are together, is

Options

A. `1/700`
B. `3/700`
C. `3/720`
D. `9/730`

Answer

To solve the problem, we need to find the probability that members of the same family are together when arranging them in a line. Let's break this down step by step.

### Step 1: Identify the Families and Their Members
We have three families:
- Family A: 3 members (let’s say A1, A2, A3)
- Family B: 3 members (let’s say B1, B2, B3)
- Family C: 4 members (let’s say C1, C2, C3, C4)

### Step 2: Calculate the Total Number of Arrangements
The total number of members is:
\[ 3 + 3 + 4 = 10 \]
The total arrangements of these 10 members without any restrictions is given by:
\[ 10! \]

### Step 3: Calculate the Arrangements with Families Together
To find the arrangements where members of the same family are together, we can treat each family as a single unit or block.

- Family A (3 members) can be arranged among themselves in:
\[ 3! \]
- Family B (3 members) can be arranged among themselves in:
\[ 3! \]
- Family C (4 members) can be arranged among themselves in:
\[ 4! \]

Now, we treat each family as a single block. Thus, we have 3 blocks (Family A, Family B, Family C) to arrange. The number of ways to arrange these 3 blocks is:
\[ 3! \]

### Step 4: Combine the Arrangements
Now, the total arrangements where members of the same family are together is given by:
\[ 3! \times 3! \times 4! \times 3! \]

### Step 5: Calculate the Probability
The probability \( P \) that members of the same family are together is given by the ratio of the favorable arrangements to the total arrangements:
\[ P = \frac{3! \times 3! \times 4! \times 3!}{10!} \]

### Step 6: Simplify the Expression
Calculating the factorials:
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 10! = 3628800 \)

Now substituting these values:
\[ P = \frac{6 \times 6 \times 24 \times 6}{3628800} \]

Calculating the numerator:
\[ 6 \times 6 = 36 \]
\[ 36 \times 24 = 864 \]
\[ 864 \times 6 = 5184 \]
So, we have:
\[ P = \frac{5184}{3628800} \]

### Step 7: Simplify the Fraction
To simplify \( \frac{5184}{3628800} \):
\[ P = \frac{1}{700} \]

### Final Answer
Thus, the probability that members of the same family are together is:
\[ \boxed{\frac{1}{700}} \]