Fundamental Principle of Counting:
If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n.
This principle can be generalized f multiple items‘ If an event can occur in m different ways, following which another event can occur in n different ways, following which a third event can occur in p different ways, then the total number of occurrence to‘ the events in the given order is m × n × p.”
Permutation with distinct objects
Let’ s assume the word“ CARD” with distinct objects C, A, R,& D. How many 4 & 2 letter words can we form using letters of CARD, when repetition of letters is not allowed, also find the number of 4 letters word if repletion is allowed?
Solution: Since we need to create 4 letters word, let’ s assume we have 4 bucket.
Case 1: 4 letters Repetition is not allowed
In this case, bucket 1 can have any of the 4 letters“ C, A, R, D”. That is 4 choices
Bucket 2 can have any of the remaining 3 letters as one letter is now inside bucket 1, that is 3 choices
Bucket 3 will can any any of the remaining 2 letters that is 2 choices Bucket 4 will have only 1 choice that is the remaining letter. Thus 1 choice.
If you multiple all these choices: 4 * 3 * 2 * 1, you will get the number of possible 4 letter words with“ C A R D”, without repetition
Thus number of 4 letter possible words are: 4 * 3 * 2 * 1 = 24 words when repetition is not allowed.
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