State and prove “ Permutations when all the objects are distinct”
Answers
Answer:
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be \displaystyle 12!12! ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the \displaystyle 12!12! permutations we counted are duplicates. The general formula for this situation is as follows.
n
!
r
1
!
r
2
!
…
r
k
!
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are \displaystyle 4!4! ways to order the stars and \displaystyle 3!3! ways to order the moon.
\displaystyle \frac{12!}{4!3!}=3\text{,}326\text{,}400
4!3!
12!
=3,326,400
There are 3,326,400 ways to order the sheet of stickers.