what is the probability of 5 people with different ages sitting in ascending or descending order at a round table.
So, let me know if there's a better way to go about this problem.
Let's have the people be named 1,2,3,4,5
They could sit:
12345
23451
34512
45123
51234
or the reverse since order matters (12345 is different from 54321)
There are 5! or 120 different ways the people can sit
So 10/120 or 1/12 is the chance that they sit in ascending or descending order.
Is there a more formal way to do this?
Also, how many ways can 5 people sit at a round table? (combination problem, order doesn't matter)
Best Answer
The answer to your second question will help illuminate a different way of thinking about the first.
If the $5$ people were to sit in a straight line, then the number of ways they can sit is indeed $5!$. Since we are asking them to sit in a circle, each possible circular arrangement corresponds to $5$ different linear arrangements, depending on which seat we label the 'first' seat. This shows that there are $5!/5=4!=24$ ways for $5$ people to sit around a table.
Now, of all $24$ possible seating arrangments, only $2$ of them are arranged in increasing or decreasing order of age, so the desired probability is $1/12$.
Notice our answers are the same, but it is important to see the distinction in our reasoning. In a sense, you've chosen a first seat, which increased your numerator and denominator by a factor of $5$. Since the question asks for a ratio, this didn't affect the final answer, but we should still note the difference between sitting around a table, and sitting in a line.