Calculating the Probability of Two Students Sitting Together

We are asked to find the probability that two specific students, P and Q, sit together in a class of \(n\) students when seats are assigned randomly.

To solve this, we need to determine two things:

1. The total number of ways the \(n\) students can be seated.
2. The number of ways P and Q can sit together.

Total Number of Seating Arrangements

If there are \(n\) students and \(n\) seats, the total number of ways to arrange these \(n\) students is the number of permutations of \(n\) distinct items, which is \(n!\) (n factorial).

Total number of outcomes = \(n!\)

Number of Arrangements Where P and Q Sit Together

To find the number of ways P and Q sit together, we can treat P and Q as a single unit or block. Now, instead of \(n\) individual students, we have (n-2) individual students plus the block containing P and Q. This gives us a total of (n-2) + 1 = (n-1) entities to arrange.

• The (n-1) entities can be arranged in \((n-1)!\) ways.
• Within the block containing P and Q, the two students can arrange themselves in \(2!\) ways (either P then Q, or Q then P). \(2! = 2\).

So, the total number of arrangements where P and Q sit together is the product of the number of ways to arrange the entities and the number of ways P and Q can arrange themselves within their block.

Number of favorable outcomes = \((n-1)! \times 2\)

Calculating the Probability

The probability that P and Q sit together is the ratio of the number of favorable outcomes (P and Q sit together) to the total number of possible outcomes (all possible seating arrangements).

Probability = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)

Probability = \(\frac{(n-1)! \times 2}{n!}\)

We know that \(n! = n \times (n-1)!\). Substituting this into the probability formula:

Probability = \(\frac{(n-1)! \times 2}{n \times (n-1)!}\)

We can cancel out the \((n-1)!\) term from the numerator and the denominator:

Probability = \(\frac{2}{n}\)

Therefore, the probability that P and Q sit together if seats are assigned randomly in a class of \(n\) students is \(\frac{2}{n}\).