1. Multiplication Principle (Fundamental Counting Principle)

The Multiplication Principle is used to find the total number of outcomes of a sequence of events.

• Rule: If an operation can be performed in $m$ ways and a second operation in $n$ ways, then the total number of ways both can occur is:

  $$\text{Total ways} = m \times n$$

• Example:

  • A shirt can be red or blue (2 options) and pants can be black or white (2 options).
  • Total outfits = $2 \times 2 = 4$

• Extension: For $k$ operations with $n_1, n_2, ..., n_k$ ways:

  $$\text{Total ways} = n_1 \times n_2 \times ... \times n_k$$

2. Permutation

Permutation refers to the arrangement of objects in a specific order. Order matters in permutation.

Formulas

1. Permutation of $n$ distinct objects (all objects):

   $$P_n = n! = n \cdot (n-1) \cdot (n-2) \cdots 1$$

2. Permutation of $r$ objects out of $n$ (ordered selection):

   $$P(n, r) = \frac{n!}{(n-r)!}$$

• Example: How many ways can 3 students be seated in 5 chairs? $$P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$$

3. Permutation with repetition (if some objects are identical): $$\text{Total arrangements} = \frac{n!}{p_1! , p_2! , ... , p_k!}$$

• Where $p_1, p_2, ..., p_k$ are counts of identical objects.
• Example: ARRANGE letters in "LEVEL" → $5!/ (2!2!) = 30$

3. Combination

Combination refers to selection of objects where order does not matter.

Formula

• Example: How many ways to choose 3 students from 5? $$C(5,3) = \frac{5!}{3!2!} = \frac{120}{6 \cdot 2} = 10$$

Relationship Between Permutation and Combination

• Because each combination of $r$ objects can be arranged in $r!$ ways.

Quick Tips to Remember

Key Points

1. Use Multiplication Principle when counting sequences of independent events.
2. Use Permutations when arrangement matters.
3. Use Combinations when only selection matters.
4. Remember: $P(n, r) = C(n, r) \cdot r!$