1. ## number of ways

2 3 4 5 6 7 8 9 }

in how many ways you can obtain 4 pairs from the following with no pair having a GCD 2

2. Hello, dhruviboy!

{ 2 3 4 5 6 7 8 9 }

In how many ways you can obtain 4 pairs with no pair having a GCD 2?
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.

3. 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.
sir you have quoted"A pair has GCD 2 if both numbers are even". then please tell what is the gcd of 4 and 8.