To determine how many non-empty subsets of the set [tex]\( A = \{1, 2, 3, 4, 5, 6\} \)[/tex] have at most 4 elements, we need to count the subsets of different sizes that meet the criterion. Let's break this down step by step:
1. Calculate the number of subsets with 1 element:
We can select 1 element from a set of 6 in [tex]\( \binom{6}{1} \)[/tex] ways.
[tex]\[
\binom{6}{1} = 6
\][/tex]
2. Calculate the number of subsets with 2 elements:
We can select 2 elements from a set of 6 in [tex]\( \binom{6}{2} \)[/tex] ways.
[tex]\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\][/tex]
3. Calculate the number of subsets with 3 elements:
We can select 3 elements from a set of 6 in [tex]\( \binom{6}{3} \)[/tex] ways.
[tex]\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\][/tex]
4. Calculate the number of subsets with 4 elements:
We can select 4 elements from a set of 6 in [tex]\( \binom{6}{4} \)[/tex] ways. Since [tex]\( \binom{6}{4} = \binom{6}{2} \)[/tex], we have:
[tex]\[
\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15
\][/tex]
Next, we sum these values to find the total number of subsets with 1, 2, 3, or 4 elements:
[tex]\[
\binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} = 6 + 15 + 20 + 15 = 56
\][/tex]
Thus, the number of non-empty subsets of set [tex]\( A \)[/tex] with at most 4 elements is:
[tex]\[
\boxed{56}
\][/tex]
