Delegates of the five of the member countries of SAARC decide to hold a round table conference. There are 2 Indians, 2 Bangladeshis, 2 Pakistanis, 2 Srilankans and 2 Nepalis. The probability that people of same nationality sit together is

Options

A. `(3!xx(2!)^(5))/(9!)`
B. `2/189`
C. `((4!)xx(2!)^(5))/(9!)`
D. `2/945`

Answer

To find the probability that delegates of the same nationality sit together at a round table, we can follow these steps:

### Step 1: Determine the total number of delegates
We have:
- 2 Indians
- 2 Bangladeshis
- 2 Pakistanis
- 2 Sri Lankans
- 2 Nepalis

Total number of delegates = 2 + 2 + 2 + 2 + 2 = 10 delegates.

### Step 2: Calculate the total arrangements around a round table
For n people sitting around a round table, the number of arrangements is given by (n - 1)!.

Here, n = 10, so the total arrangements = (10 - 1)! = 9! = 362880.

### Step 3: Group delegates of the same nationality
If we consider each pair of delegates from the same country as a single unit, we can treat them as 5 units:
- 1 unit for Indians (2 Indians)
- 1 unit for Bangladeshis (2 Bangladeshis)
- 1 unit for Pakistanis (2 Pakistanis)
- 1 unit for Sri Lankans (2 Sri Lankans)
- 1 unit for Nepalis (2 Nepalis)

### Step 4: Calculate arrangements of these units around the table
Now, we have 5 units to arrange around the table. The number of arrangements for these 5 units is (5 - 1)! = 4! = 24.

### Step 5: Arrange delegates within each unit
Within each unit, the 2 delegates can be arranged in 2! ways. Since there are 5 units, the total arrangements within the units will be (2!)^5 = 2^5 = 32.

### Step 6: Calculate the total favorable arrangements
The total favorable arrangements where delegates of the same nationality sit together is given by:
Total favorable arrangements = arrangements of units × arrangements within units
= 4! × (2!)^5
= 24 × 32
= 768.

### Step 7: Calculate the probability
The probability that delegates of the same nationality sit together is given by:
Probability = (Total favorable arrangements) / (Total arrangements)
= 768 / 362880.

### Step 8: Simplify the probability
To simplify:
768 / 362880 = 1 / 472.5 (approximately).

Thus, the final probability that delegates of the same nationality sit together is approximately 1/472.5.

### Final Answer
The probability that people of the same nationality sit together is approximately 1/472.5.

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