A Fundamental Principal of Counting - Permutations and Combinations
Fundamental principal of counting (F.P.C)
The fundamental principle of counting is a basic rule in mathematics that helps us determine the total number of possible outcomes in a sequence of events. In simple words, it says that if we have several choices to make, and each choice is independent of the others, we can find the total number of combinations by multiplying the number of choices at each step.
For example:If we have 3 shirts and 2 pairs of pants, we can create 3 × 2 = 6 different outfits.
Addition Law in Fundamental Principle of Counting
If there are two operations such that they can be performed independently in m and n ways respectively, then either of the two operations can be performed in m + n ways.
Example: We have two different activities: reading a book or watching a movie.
- There are 4 different books we can read.
- There are 3 different movies we can watch.
Since we can choose to either read a book or watch a movie, but not both at the same time, we apply the addition law.
Application of Addition Law: According to the addition law, the total number of ways to perform either of the two activities (reading a book or watching a movie) is:
4 (ways to read a book) + 3 (ways to watch a movie) = 7 (total ways to choose an activity)
So, we have 7 different ways to either read a book or watch a movie.
The addition law helps us calculate the total number of possible outcomes when choosing between distinct and non-overlapping events. In this example, we have 7 ways to choose between reading a book or watching a movie.
Multiplication Law in Fundamental Principle of Counting
If there are two operations such that the first operation can be performed in m ways, and the second operation can be performed in n ways, then the total number of ways to perform both operations in sequence is m × n.
Example: We have two different activities: choosing a shirt and choosing a pair of pants.
- There are 3 different shirts we can choose from.
- There are 2 different pairs of pants we can choose from.
Since we can choose a shirt and a pair of pants independently, we apply the multiplication law.
Application of Multiplication Law: According to the multiplication law, the total number of ways to choose both a shirt and a pair of pants is:
3 (ways to choose a shirt) × 2 (ways to choose a pair of pants) = 6 (total combinations)
So, we have 6 different ways to choose an outfit consisting of a shirt and a pair of pants.
The multiplication law helps us calculate the total number of possible outcomes when performing a sequence of independent operations. In this example, we have 6 ways to choose an outfit by combining a shirt and a pair of pants.
Practice Questions.
Question 1. How many 5 digits telephone number can be constructed using the digits 0 to 9 if each number starts with 67, for example 67125 etc., and no digit appears more than once?
Solution:
According to the problem two digit 67 are fixed. Thus, 10-2=8 digit can be used in contructing the telephone numbers. There are 8 digit 0,1,2,2,4,5,8,9
First number can be selected in 8 ways.
After the selection of first digit we have 8-1=7 digits in hand, the second digit can be selected n 7 ways. and the third digit can be selected in 6 ways.
According to F.P.C. number of ways of selecting the digits = 8 x 7 x 6 = 336.
Question 2. A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many many possible outomces if the coin is tossed four times? Five times? n times?
Solution:
In tossing a coint there are two possible outcomes, H or T (i.e., Head or Tail). Similarly, tossing the coin for the second time, there are again the same two outcomes and similiarly for third toss the same outcomes appear. Total number of outcomes in three tosses: 2 x 2 x 2 = 23
If the coin tossed four times, the number of outcomes = 2 x 2 x 2 x 2 = 24
If the coin is tossed for five time, then the number of outcomes = 25 ways and if the coin is tossed for n time, then the number of outcomes = 2 n ways.
Hence the number of outcomes are 23, 24, 25 and 2n
