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A combinatorics problem: John has 5 pairs of jeans, 12 different shirts, and 4 pairs of shoes. How many different outfits can he create? Complete solution using the fundamental counting principle with clear explanation.

Problem

John has five pairs of jeans, 12 different shirts, and 4 pairs of shoes. How many different outfit combinations can he create?

Key Concepts

  • Multiplication Principle: When an event has multiple independent stages, the total number of outcomes is the product of the number of outcomes at each stage.
  • Independent Selection: The choice of one item does not affect the availability of items in other categories.
  • Complete Outfit: Consists of exactly one item from each category (one pair of jeans, one shirt, one pair of shoes).

Solution

👉 View solution

This problem can be solved using the Multiplication Principle (also known as the Fundamental Counting Principle).

Since John needs to select:

  • 1 pair of jeans from 5 options
  • 1 shirt from 12 options
  • 1 pair of shoes from 4 options

And these selections are independent of each other (choosing one pair of jeans doesn’t affect which shirt or shoes he can choose), we multiply the number of options for each item:

Total number of outfits=Number of jeans×Number of shirts×Number of shoes\text{Total number of outfits} = \text{Number of jeans} \times \text{Number of shirts} \times \text{Number of shoes} Total number of outfits=5×12×4=240\text{Total number of outfits} = 5 \times 12 \times 4 = 240

Therefore, John can create 240 different outfit combinations with his wardrobe.

Applications

The Multiplication Principle is widely used in:

  • Fashion and wardrobe planning
  • Menu creation (selecting one item from each course)
  • Travel planning (different combinations of transportation, accommodations, activities)
  • Product configurations (selecting options for a customizable product)
  • Password and code possibilities
  • Addition Principle: Used when counting the total outcomes from multiple exclusive events
  • Permutations: Used when order matters in selections
  • Combinations: Used when selecting multiple items from the same category without regard to order
# Statistics # Mathematics # Probability # Combinatorics # Counting Principles

Frequently Asked Questions

What is the Multiplication Principle in combinatorics? ▼

The Multiplication Principle (also called the Fundamental Counting Principle) states that if one event can occur in m ways, and a second event can occur in n ways, then the two events in sequence can occur in m × n ways.

How do you apply the Multiplication Principle to multiple items? ▼

When there are more than two categories, you simply multiply all the possibilities together. For example, with three categories having a, b, and c possibilities respectively, the total number of combinations is a × b × c.

What's the difference between combinations and the Multiplication Principle? ▼

The Multiplication Principle is used when you're selecting one item from each category, while combinations are used when selecting multiple items from a single category without regard to order.