GATE CS 2007 Q24 — Probability: Random Variables and Distributions

To find the probability that $2$ appears at an earlier position than any other even number, we can focus solely on the relative ordering of the even numbers in the permutation.

The set of even numbers in the range $[1, 20]$ is:

There are exactly $10$ even numbers in this set.

In any randomly and uniformly chosen permutation of the numbers $1, 2, \dots, 20$, all possible relative orderings of these $10$ even numbers are equally likely.

• The total number of ways to arrange these $10$ even numbers among themselves is $10!$.
• For $2$ to appear earlier than any other even number, $2$ must be the first element in this relative sequence of even numbers.
• The remaining $9$ even numbers can be arranged in any of the $9!$ ways behind $2$.

Therefore, the probability that $2$ appears first among all the even numbers is: