Permutations and Combinations Ex 7.1: Fundamental Principle of Counting

The Fundamental Principle of Counting: Multiplication and Addition

The Fundamental Principle of Counting (FPC) is the bedrock of combinatorics. It provides a systematic way to count the total number of outcomes when multiple independent events occur. There are two main parts to the FPC:

1. Fundamental Principle of Multiplication: 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 both events can occur in a definite order is m × n. This principle can be extended to any finite number of events. For example, if you have 3 shirts and 2 trousers, the number of different outfits you can make is 3 × 2 = 6. You choose a shirt AND then choose trousers. The keyword here is 'AND' – events happen sequentially.

2. Fundamental Principle of Addition: If an event can occur in 'm' different ways, and another independent event, which is mutually exclusive to the first (meaning they cannot happen at the same time), can occur in 'n' different ways, then the total number of ways that either of these events can occur is m + n. For example, if you can choose a book from 5 different math books OR 3 different science books, the total number of choices is 5 + 3 = 8. The keyword here is 'OR' – you choose one event or the other. While Exercise 7.1 primarily focuses on the multiplication principle for arranging distinct objects, understanding both aspects of FPC is essential for future problems.

Key Definitions

Event

An operation or task that can be performed, which has several possible outcomes.

Outcome

A single result of an experiment or event.

Fundamental Principle of Counting (FPC)

A basic counting technique used to determine the total number of possible outcomes for a sequence of events. It comprises the multiplication and addition principles.

Permutation

An arrangement of objects in a specific order. Order matters in permutations.

Combination

A selection of objects where the order does not matter. It's about choosing items from a group.

Worked Examples from NCERT Exercise 7.1

• Example 1: Forming 3-digit numbers without repetition How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 assuming that repetition of the digits is not allowed? Step 1: Identify the positions to fill. We need to form a 3-digit number, so there are three positions: hundreds, tens, and units place. Step 2: Determine choices for the first position (hundreds place). Since repetition is not allowed, for the hundreds place, we have 5 choices (any of 1, 2, 3, 4, 5). Step 3: Determine choices for the second position (tens place). After filling the hundreds place, one digit is used. So, for the tens place, we are left with 4 choices from the remaining digits. Step 4: Determine choices for the third position (units place). After filling the hundreds and tens places, two digits are used. So, for the units place, we are left with 3 choices from the remaining digits. Step 5: Apply the Fundamental Principle of Multiplication. The total number of ways to form the 3-digit number is the product of the number of choices for each position: 5 × 4 × 3 = 60. Final Answer: 60 different 3-digit numbers can be formed.
• Example 2: Forming 3-digit even numbers with repetition How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated? Step 1: Identify the positions and constraints. We need a 3-digit even number, meaning the units place must be an even digit. Digits can be repeated. Step 2: Start with the constrained position (units place). For the number to be even, the units digit must be 2, 4, or 6. So, there are 3 choices for the units place. Step 3: Determine choices for the first position (hundreds place). Since repetition is allowed, any of the 6 digits (1, 2, 3, 4, 5, 6) can be used for the hundreds place. So, there are 6 choices. Step 4: Determine choices for the second position (tens place). Similarly, since repetition is allowed, any of the 6 digits can be used for the tens place. So, there are 6 choices. Step 5: Apply the Fundamental Principle of Multiplication. The total number of ways to form the 3-digit even number is: 6 × 6 × 3 = 108. Final Answer: 108 different 3-digit even numbers can be formed.
• Example 3: Forming codes with letters and numbers A code consists of two distinct letters followed by two distinct digits. How many such codes are possible? Step 1: Break down the problem into events. We need to choose the first letter, then the second letter, then the first digit, and finally the second digit. Step 2: Determine choices for the first letter. There are 26 possible letters in the English alphabet. So, 26 choices for the first letter. Step 3: Determine choices for the second letter. Since the letters must be distinct, one letter is already used. So, there are 25 choices remaining for the second letter. Step 4: Determine choices for the first digit. There are 10 possible digits (0, 1, 2, ..., 9). So, 10 choices for the first digit. Step 5: Determine choices for the second digit. Since the digits must be distinct, one digit is already used. So, there are 9 choices remaining for the second digit. Step 6: Apply the Fundamental Principle of Multiplication. Total number of codes = (Choices for 1st letter) × (Choices for 2nd letter) × (Choices for 1st digit) × (Choices for 2nd digit) = 26 × 25 × 10 × 9 = 58,500. Final Answer: 58,500 such codes are possible.

Exam Tips and Common Mistakes

When tackling problems based on the Fundamental Principle of Counting, especially from Exercise 7.1, keep these points in mind:

1. Read Carefully for Repetition: The most common mistake is misinterpreting whether repetition is allowed or not. If a problem states "without repetition" or "distinct," the number of choices for subsequent positions will decrease. If it says "with repetition" or doesn't specify, assume repetition is allowed.
2. Start with Constraints: If there's a specific condition (e.g., "even number," "number greater than 500"), always start filling the position that is affected by this constraint first. For even numbers, start with the units place. For numbers greater than 500, start with the hundreds place.
3. Use the 'Boxes' Method: Visualize each position as a box. Write the number of choices available for each box. Then, multiply the numbers in the boxes. This systematic approach helps prevent errors.
4. Distinguish 'AND' vs. 'OR': Remember, the multiplication principle applies when events occur consecutively (Event A AND Event B). The addition principle applies when you choose one of several mutually exclusive options (Event A OR Event B). Most problems in Ex 7.1 involve sequential choices, thus the multiplication principle.
5. Be Mindful of Zero: If 0 is one of the available digits, remember that it cannot be in the leading position of a number (e.g., 012 is not a 3-digit number). This creates an additional constraint for the first position.

Practice Questions with Solutions

• Q: How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated? A: Step 1: Identify the number of positions to fill (4 letters). Step 2: For the first position, there are 10 choices (A-J). Step 3: For the second position, since repetition is not allowed, there are 9 remaining choices. Step 4: For the third position, there are 8 remaining choices. Step 5: For the fourth position, there are 7 remaining choices. Step 6: Apply the multiplication principle: 10 × 9 × 8 × 7 = 5040. Final answer: 5040 four-letter codes can be formed.
• Q: A coin is tossed 3 times. What is the total number of possible outcomes? A: Step 1: Identify the number of events (3 tosses). Step 2: For the first toss, there are 2 possible outcomes (Head or Tail). Step 3: For the second toss, there are still 2 possible outcomes (independent event). Step 4: For the third toss, there are also 2 possible outcomes. Step 5: Apply the multiplication principle: 2 × 2 × 2 = 8. Final answer: There are 8 possible outcomes.
• Q: How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, if repetition of digits is allowed? A: Step 1: Identify the positions to fill (hundreds, tens, units). Step 2: For the hundreds place, 0 cannot be used. So, there are 5 choices (1, 2, 3, 4, 5). Step 3: For the tens place, repetition is allowed, so all 6 digits (0, 1, 2, 3, 4, 5) are available. So, 6 choices. Step 4: For the units place, repetition is allowed, so all 6 digits are available. So, 6 choices. Step 5: Apply the multiplication principle: 5 × 6 × 6 = 180. Final answer: 180 three-digit numbers can be formed.
• Q: In a local election, there are 3 candidates for President, 2 for Vice-President, and 4 for Secretary. In how many ways can these three offices be filled? A: Step 1: Identify the independent events (choosing a President, a Vice-President, and a Secretary). Step 2: The number of ways to choose a President is 3. Step 3: The number of ways to choose a Vice-President is 2. Step 4: The number of ways to choose a Secretary is 4. Step 5: Apply the multiplication principle: 3 × 2 × 4 = 24. Final answer: The three offices can be filled in 24 ways.

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