Originally Posted by
Soroban Hello, dhruviboy!
A pair has GCD 2 if both numbers are even.
. . Hence, we must form pairs in which one is odd and one is even.
We have the four odd numbers: .$\displaystyle 3\;\;5\;\;7\;\;9$
In how many ways can the even numbers be assigned?
The "2" can be paired with any of the 4 odd numbers.
Then the "4" can be paired with any of the remaining 3 odd numbers.
Then the "6" can be paired with either of the remaining 2 odd numbers.
And the "8" is paired with the 1 remaining odd number.
Therefore, there are: .$\displaystyle 4! \:=\:{\color{blue}24}$ possible pairings.