Problem 6

Explanation

The order of members in a set does not matter, so we use the concept of combinations.

The total number of subsets of a set with $n$ members is $2^n$. Given $n = 10$, the total number of subsets is:

We are asked to find the number of subsets with more than $3$ members. This means we are looking for the sum of subsets with 4, 5, 6, ..., 10 members.

An easier way is to use the complement principle: Total subsets minus subsets with $0$, $1$, $2$, or $3$ members.

The combination formula is:

Let's calculate them one by one.

Subsets with 0 members

Subsets with 1 member

Subsets with 2 members

Subsets with 3 members

The number of subsets with $\leq 3$ members is:

Thus, the number of subsets with more than $3$ members is:

So, the number of subsets of $A$ that have more than $3$ members is $848$.