How many subsets of cardinality 3 does A have if |A| = n?
Understand the Problem
The question is asking for the number of subsets that can be formed from a set A that contains n elements, specifically, we need to find the number of subsets that have exactly 3 elements.
Answer
The number of subsets with exactly 3 elements from a set of $n$ elements is given by $$ C(n, 3) = \frac{n(n-1)(n-2)}{6} $$
Answer for screen readers
The number of subsets with exactly 3 elements from a set of $n$ elements is given by
$$ C(n, 3) = \frac{n(n-1)(n-2)}{6} $$
Steps to Solve
- Identify the Set Size
The set A has $n$ elements. We need to find the number of subsets with exactly 3 elements.
- Use the Combination Formula
To calculate the number of ways to choose 3 elements from a set of $n$ elements, we use the combination formula:
$$ C(n, k) = \frac{n!}{k!(n-k)!} $$
Here, $k = 3$. So the formula becomes:
$$ C(n, 3) = \frac{n!}{3!(n-3)!} $$
- Simplify the Combination Formula
Since $3! = 6$, we can rewrite the combination formula as:
$$ C(n, 3) = \frac{n(n-1)(n-2)}{6} $$
This formula gives us the number of subsets of size 3.
The number of subsets with exactly 3 elements from a set of $n$ elements is given by
$$ C(n, 3) = \frac{n(n-1)(n-2)}{6} $$
More Information
This calculation assumes that the elements of set A are distinct. The formula counts all unique combinations of 3 elements that can be drawn from the set. This concept is crucial in combinatorics and has applications in various fields, including probability and statistics.
Tips
- Confusing combinations with permutations: Remember that the order in which you choose elements does not matter in combinations. Always use the combination formula when selecting groups where order is not important.
- Forgetting to check if $n \geq 3$: It's important to verify that the number of elements in the set is at least 3. If $n < 3$, the number of subsets with 3 elements would be zero.
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