Fundamental Principle of Counting

Definition: The Fundamental Principle of Counting (FPC) is a way to figure out the number of possible outcomes for a given situation. It is a basic principle that helps us to count large numbers in a non-tedious way. The multiplication principle states that if an event A can occur in x different ways and another event B can occur in y different ways, then there are x × y ways of occurrence of both the events simultaneously. This principle can be used to predict the number of ways of occurrence of any number of finite events. For example, if there are 4 events which can occur in p, q, r, and s ways, then there are p × q × r × s ways in which these events can occur simultaneously.

Key Concepts:

  1. Fundamental Principle of Counting (FPC): The FPC states that if there are "m" ways to do one thing and "n" ways to do another, then there are "m x n" ways to do both things together. In other words, the total number of outcomes for a sequence of events is the product of the number of outcomes for each event.

  2. Independent Events: The FPC is most commonly applied to independent events. Independent events are events in which the outcome of one event does not affect the outcome of another. For example, when rolling a die and flipping a coin, these events are independent because the result of rolling the die does not impact the outcome of flipping the coin.

  3. Example 1: Rolling a Die and Flipping a Coin Suppose you want to count the number of possible outcomes when rolling a standard six-sided die and flipping a coin. The FPC can be applied here.

    • There are 6 possible outcomes when rolling the die (1, 2, 3, 4, 5, 6).
    • There are 2 possible outcomes when flipping the coin (Heads or Tails).

    To find the total number of outcomes when both events occur, apply the FPC: 6 (die outcomes) x 2 (coin outcomes) = 12 possible outcomes.

  4. Example 2: Selecting an Outfit Let's say you want to count the number of possible outfits you can create from a selection of 4 shirts and 3 pairs of pants.

    • There are 4 ways to choose a shirt.
    • There are 3 ways to choose a pair of pants.

    Using the FPC, you can calculate the total number of outfits: 4 (shirt choices) x 3 (pants choices) = 12 possible outfits.

  5. Example 3: Suppose you have 3 pairs of shoes and 4 pairs of socks. In how many ways can you wear them? So, we can wear it 3 × 4 ways, i.e., 12 ways
  6. Example 4: In how many ways a 4 digit pin can be set up without repetition of digits? The number of ways to set up a 4 digit pin without repetition of digits is 10 × 9 × 8 × 7 = 5040.
  7. Example 5: In a Felicitation Ceremony, in how many ways two guests can shake hands with each other and top three rank holders? The number of ways two guests can shake hands with each other and top three rank holders is 2 × 3 = 6.
  8. Example 6: In how many ways a six-digit PIN Code can be created? The number of ways a six-digit PIN Code can be created is 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000.
  9. Multiple Stages: The FPC can be extended to situations with more than two events. If you have a sequence of events, you can use the FPC for each stage and then multiply the results to find the total number of outcomes.

  10. Permutations and Combinations: The FPC is a fundamental principle that forms the basis for calculating permutations and combinations, which are used to determine the number of arrangements and selections of items from a set.

  11. Application in Probability: The FPC is widely used in probability theory to calculate the probabilities of various outcomes when multiple events occur independently. It is a crucial tool for solving problems related to probability and statistics.