Math Problem Statement
Solution
The problem asks how many zero-element subsets can be chosen from a two-element set, using Pascal's triangle to find the answer.
Solution:
-
Pascal's Triangle:
- The rows in Pascal's triangle are indexed starting from .
- Row 2 of Pascal’s triangle corresponds to and is:
The elements in this row represent the binomial coefficients , where ranges from to .
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Subset of size 0:
- The number of subsets of size 0 is given by .
- For , .
This means there is exactly 1 subset of size 0, which is the empty set .
Final Answer:
The number of zero-element subsets of a two-element set is 1.
Would you like me to explain Pascal's triangle further or clarify any concepts?
Here are some follow-up questions to expand on this:
- What are the values of for and ?
- How do the rows of Pascal’s triangle correspond to subsets of a set?
- Can you explain why is always equal to 1 for any ?
- How does Pascal's triangle relate to the binomial theorem?
- How would the solution change if the set had 3 elements instead of 2?
Tip: The top row of Pascal's triangle () always corresponds to the case where the set is empty.
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Pascal's Triangle
Subsets
Formulas
Binomial Coefficient Formula: C(n, k) = n! / (k!(n-k)!)
Theorems
Properties of Pascal's Triangle
Suitable Grade Level
Grades 6-8
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