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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