Problem: Find the number of good subsets of the set {2,3,4,5,6,7,8,9,10}, where a good subset is a subset with exactly 4 elements and their gcd is 1.

Solution:

To find the number of good subsets, we need to first find the number of ways we can choose 4 elements from the given set. This can be done using the combination formula:

nCr = n! / (r! * (n-r)!)

where n is the total number of elements in the set and r is the number of elements we want to choose.

Substituting the values, we get:

9C4 = 9! / (4! * 5!) = 126

So, there are 126 ways to choose 4 elements from the given set.

Now we need to count the number of subsets with gcd 1. To do this, we can use the principle of inclusion-exclusion.

Let A be the set of all subsets with gcd divisible by 2, B be the set of all subsets with gcd divisible by 3, C be the set of all subsets with gcd divisible by 5, and D be the set of all subsets with gcd divisible by 7.

Then, the number of good subsets is given by:

|U| - |A ∪ B ∪ C ∪ D|

where |U| is the total number of subsets and |A ∪ B ∪ C ∪ D| is the number of subsets that have gcd divisible by at least one of 2, 3, 5, 7.

We can find |U| using the formula 2^n, where n is the total number of elements in the set. Substituting, we get:

|U| = 2^9 = 512

To find |A ∪ B ∪ C ∪ D|, we need to find the number of subsets with gcd divisible by 2, 3, 5, and 7 separately, and then subtract the subsets with gcd divisible by the intersection of any two, three, or all four of these.

For example, to find the number of subsets with gcd divisible by 2, we can count the number of subsets that contain only even numbers (2, 4, 6, 8, 10) and subtract the subsets that have gcd divisible by 4 (i.e. subsets that contain 4).

Similarly, to find the number of subsets with gcd divisible by 3, we can count the number of subsets that contain only multiples of 3 (3, 6, 9) and subtract the subsets that have gcd divisible by 9 (i.e. subsets that contain 9).

Using this method, we can find the number of subsets with gcd divisible by 5 and 7 as well.

Finally, we need to subtract the subsets with gcd divisible by the intersection of any two, three, or all four of these. For example, to find the number of subsets with gcd divisible by both 2 and 3, we can count the number of subsets that contain only multiples of 6 (6) and subtract the subsets that have gcd divisible by 6 (i.e. subsets that contain both 2 and 3).

Using this method, we can find the number of subsets with gcd divisible by any two