Introduction

Permutations and Combinations is the study of counting arrangements and selections of objects. The entire field is built upon a simple but powerful idea known as the Fundamental Principle of Counting.

1. Multiplication Principle (The 'AND' Rule)

This principle states that if an event can occur in m different ways, and following it, a second event can occur in n different ways, then the total number of ways that the two events can occur in succession is m × n.

This principle can be extended to any finite number of events. If a sequence of tasks T1,T2,...,TkT_1, T_2, ..., T_k can be performed in n1,n2,...,nkn_1, n_2, ..., n_k ways respectively, then the total number of ways of performing all the tasks is n1×n2×...×nkn_1 \times n_2 \times ... \times n_k.

  • Keyword: Think 'AND'. If you have to do Task 1 AND Task 2, you multiply the number of ways.

2. Addition Principle (The 'OR' Rule)

This principle states that if there are two independent jobs, and the first can be performed in m ways and the second can be performed in n ways, then either of the two jobs can be performed in m + n ways.

This applies when the two tasks are mutually exclusive (i.e., you can only do one of them).

  • Keyword: Think 'OR'. If you have to do either Task 1 OR Task 2, you add the number of ways.

Example 1: Forming Numbers with Repetition

Question: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 assuming that repetition of the digits is allowed?

Solution: We have to fill three places: Units, Tens, and Hundreds.

  • The hundreds place can be filled in 5 ways (using 1, 2, 3, 4, or 5).
  • Since repetition is allowed, the tens place can also be filled in 5 ways.
  • Similarly, the units place can be filled in 5 ways. By the Multiplication Principle, the total number of 3-digit numbers is 5×5×5=1255 \times 5 \times 5 = 125.

Example 2: Forming Numbers without Repetition

Question: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 assuming that repetition of the digits is not allowed?

Solution:

  • The hundreds place can be filled in 5 ways.
  • Since repetition is not allowed, after filling the hundreds place, we have 4 digits remaining. So, the tens place can be filled in 4 ways.
  • After filling the first two places, we have 3 digits remaining. So, the units place can be filled in 3 ways. By the Multiplication Principle, the total number of 3-digit numbers is 5×4×3=605 \times 4 \times 3 = 60.

Example 3: Combined Principles with Constraints

Question: How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Solution: We always handle the constraint first. For the number to be even, the units place must be an even digit.

  • Units place: Can be filled by 2, 4, or 6. So, there are 3 ways.
  • Tens place: Repetition is allowed, so it can be filled by any of the 6 digits. There are 6 ways.
  • Hundreds place: Repetition is allowed, so it can also be filled in 6 ways. The total number of 3-digit even numbers is 6×6×3=1086 \times 6 \times 3 = 108.

Example 4: Addition Principle

Question: In a class, there are 10 girls and 8 boys. The teacher wants to select one student to represent the class. In how many ways can the teacher make this selection?

Solution: The teacher can either select a girl OR a boy.

  • The number of ways to select a girl is 10.
  • The number of ways to select a boy is 8. By the Addition Principle, the total number of ways to select one student is 10+8=1810 + 8 = 18.