When all the objects are not distinct objects Permutations When all the objects are not distinct objects

Theorem 1: The number of permutations of n objects, where p objects are of the same kind and rest are all different = Theorem 2: The number of permutations of n objects, where p1 objects are of one kind , p2 are of second kind, ..., pk are of kth kind and the rest, if any, are of different kind is

Example 1: Find the number of permutations of the letters of the word COCONUT. Solution: Here, there are 7 objects (letters) of which there are 2C’s, 2 O’s and rest are all different. Therefore, the required number of arrangements: Hence, there are 1260 permutations of the letters of the word COCONUT.

Example 2: In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together? Solution: ABACUS is a 6 letter word with 3 of the letters being vowels. If the 3 vowels have to appear together, then there will 3 other consonants and a set of 3 vowels together. These 4 elements can be rearranged in 4! Ways. The 3 vowels can rearrange amongst themselves in ways as "a" appears twice. Hence, the total number of rearrangements in which the vowels appear together are