To find the number of different subsets of a set of 3 elements, we can use Pascal's triangle.

Pascal's triangle helps us calculate binomial coefficients, which represent the number of subsets of a given size. The number of subsets of size $k$ of a set with $n$ elements is given by $\binom{n}{k}$, the binomial coefficient.

For a set of 3 elements, the row 3 of Pascal's triangle provides the coefficients for subsets of sizes 0, 1, 2, and 3. Here’s how Pascal’s triangle looks for the first few rows:

& & 1 & & \\ & 1 & & 1 & \\ 1 & & 2 & & 1 \\ 1 & 3 & 3 & 1 \\ \end{array}$$ In row 3 (the 4th row of Pascal's triangle), the values are: $$1, 3, 3, 1$$ These correspond to the number of subsets of size 0, 1, 2, and 3 respectively for a set of 3 elements: - $$\binom{3}{0} = 1$$ (The empty set) - $$\binom{3}{1} = 3$$ (Subsets with 1 element) - $$\binom{3}{2} = 3$$ (Subsets with 2 elements) - $$\binom{3}{3} = 1$$ (The entire set) To find the total number of subsets of the set, we sum the values in the row: $$1 + 3 + 3 + 1 = 8$$ Thus, the total number of different subsets of a set of 3 elements is **8**. Would you like to see more details or explanations? Here are some related questions you could explore: 1. How do you calculate binomial coefficients manually? 2. How does the concept of subsets relate to combinations in combinatorics? 3. Can Pascal's triangle be used to calculate combinations for larger sets? 4. What is the power set of a set with 3 elements? 5. How can you generalize the method to find the number of subsets for a set of $$n$$ elements? **Tip**: The number of subsets of a set with $$n$$ elements is always $$2^n$$, and this result is reflected in Pascal's triangle.