Math Problem Statement

How many zero-element subsets can we choose from a two-element set? (Use row 2 of Pascal's triangle to find the answer.)

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:

  1. Pascal's Triangle:

    • The rows in Pascal's triangle are indexed starting from n=0n = 0.
    • Row 2 of Pascal’s triangle corresponds to n=2n = 2 and is:
      1211 \quad 2 \quad 1 The elements in this row represent the binomial coefficients (2k)\binom{2}{k}, where kk ranges from 00 to 22.
  2. Subset of size 0:

    • The number of subsets of size 0 is given by (n0)\binom{n}{0}.
    • For n=2n = 2, (20)=1\binom{2}{0} = 1.
      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:

  1. What are the values of (2k)\binom{2}{k} for k=1k = 1 and k=2k = 2?
  2. How do the rows of Pascal’s triangle correspond to subsets of a set?
  3. Can you explain why (n0)\binom{n}{0} is always equal to 1 for any nn?
  4. How does Pascal's triangle relate to the binomial theorem?
  5. How would the solution change if the set had 3 elements instead of 2?

Tip: The top row of Pascal's triangle (n=0n = 0) 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