Question

The captains of five cricket teams, including India and Australia, are lined up randomly next to one other for a group photo. What is the probability that the captains of India and Australia will stand next to each other?

• A. \( \frac{1}{4} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{1}{5} \)

Hint:

When calculating probabilities involving arrangements, treat groups as single blocks to simplify the problem.

Answer 1

The Correct Option is B

We are given five cricket captains, including the captains of India and Australia, and need to determine the probability that they will stand next to each other.

Step 1: Total number of arrangements.
The total number of ways to arrange 5 captains is \( 5! \). Thus, the total arrangements are: \[ 5! = 120 \]

Step 2: Favorable arrangements.
Now, treat the captains of India and Australia as a block. This reduces the problem to arranging 4 blocks (the India-Australia block and the other 3 captains). The number of ways to arrange these 4 blocks is \( 4! \), and since India and Australia can be arranged within their block in \( 2! \) ways, the total favorable arrangements are: \[ 4! \times 2! = 24 \times 2 = 48 \]

Step 3: Probability.
The probability that India and Australia stand next to each other is the ratio of favorable arrangements to total arrangements: \[ P = \frac{48}{120} = \frac{2}{5} \] Thus, the correct answer is \( \frac{2}{5} \), corresponding to option (b).