Question

if sam, tim and 10 other people are randomly positioned in a queue, then what is the probability that sam and tim will be positioned next to each other?

Answer 1

To find the probability that Sam and Tim will be positioned next to each other in a randomly positioned queue, we need to consider the total number of possible permutations of the people in the queue. The probability is 1/12.

Step-by-step explanation:

To find the probability that Sam and Tim will be positioned next to each other in a randomly positioned queue with Sam, Tim, and 10 other people, we need to consider the total number of possible permutations of the people in the queue.

Since Sam and Tim need to be positioned next to each other, we can treat them as a single entity. So, we have 12 entities (Sam and Tim treated as one entity and the other 10 people) that need to be arranged in a queue.

The total number of possible permutations is 12!, which is equal to 479,001,600. Now, Sam and Tim can be arranged internally within their entity in 2! ways, which is equal to 2. So, the total number of favorable permutations is 2 x 11! (since Sam and Tim can be treated as one entity) which is equal to 39,916,800. Finally, we can find the probability by dividing the number of favorable permutations by the total number of possible permutations: 39,916,800 / 479,001,600 = 1/12

Answer 2

The probability that Sam and Tim will be positioned next to each other in the queue can be calculated using the formula P(Sam and Tim next to each other) = (12! * 2!) / 13!.

Step-by-step explanation:

To calculate the probability that Sam and Tim will be positioned next to each other in a queue, we first need to find the total number of possible positions for Sam and Tim. Since there are 12 other people in the queue, there are 13 total positions for Sam and Tim.

If we consider Sam and Tim as a single entity, there are 11 other entities in the queue. We can arrange these entities (including the Sam and Tim entity) in 12! ways. However, within the Sam and Tim entity, there are 2! ways to arrange them. So the total number of arrangements is 12! * 2!.

The probability of Sam and Tim being positioned next to each other is then the number of favorable outcomes (the number of arrangements where Sam and Tim are next to each other) divided by the total number of possible outcomes. The probability is:

P(Sam and Tim next to each other) = (12! * 2!) / 13!

Simplifying this expression will give you the exact probability.