Question

If six persons sit around a table, the probability that a certain three of them are always together is:

Answer 1

There are six persons and three of them are grouped together. Since it is a circle, this can be done in 3! 3! ways. 

There are six persons and three of them are grouped together. Since it is a circle, this can be done in 3! 3! ways. Therefore, P = 3!3!/5!

There are six persons and three of them are grouped together. Since it is a circle, this can be done in 3! 3! ways. Therefore, P = 3!3!/5!= 3/10

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Answer 2

Concept: Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about how probable an event is to happen, or its chance of happening. Probability can range from 0 to 1, with 0 denoting an impossibility and 1 denoting a certainty. For pupils in Class 10, probability is a crucial subject because it teaches all the fundamental ideas of the subject. One is the probability of every event in a sample space.

Given: six persons sit around a table

To find: the probability that a certain three of them are always together

Solution:

6 persons can sit around a table in 5! ways

Considering the three persons sitting together as 1 unit

Now the entities are 3 + 1 = 4

These 4 entities can be arranged in 3! ways

In the entities itself they can be arranged in 3! ways

The required number of arrangements = 3! 3!

$Probability = \frac{m}{n} = \frac{3! 3!}{5!} = \frac{3*2*1*3*2*1}{5*4*3*2*1} = \frac{6}{20} = \frac{3}{10}$

Hence, the probability that a certain three of them are always together is $\frac{3}{10}$.

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