Understanding Counting in Algebra
Counting principles help determine the number of ways events can occur. They are foundational in probability and combinatorics.
Fundamental Principle of Counting
If one event can occur in \( m \) ways and another in \( n \) ways, both can occur in \( m \times n \) ways.
Permutations
Permutations count the number of ways to arrange objects where order matters.
Formula: \( n! \) for arranging \( n \) objects.
Combinations
Combinations count the number of ways to select objects where order does not matter.
Formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
Practice Problems
- How many ways can 3 books be arranged on a shelf?
- How many ways can you choose 2 students from 5?
- How many 3-digit numbers can be formed using 1, 2, 3, 4 without repetition?
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