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

2. 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

3. 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.

4. 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