parveen110 wrote:In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:
If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]
Okay, to solve the question, we must recognize the following:
# of "words" where the 4 I's are NOT together = (total # of arrangements) - (# of arrangements where the 4 I's ARE together)
total # of arrangements
Here, we're dealing with the letters M, I, I, I, I, S, S, S, S, P, and P
There are
11 letters in total
There are
4 identical I's
There are
4 identical S's
There are
2 identical P's
So, the total number of possible arrangements =
11!/[(4!)(4!)(2!)]
=
69,300
# of arrangements where the 4 I's ARE together
To ensure that all 4 I's are together, let's GLUE them together to form just ONE GIANT "LETTER" (IIII). Our letters are now as follows: M, S, S, S, S, P, P, and IIII
There are
8 letters in total
There are
4 identical S's
There are
2 identical P's
So, the total number of possible arrangements =
8!/[(4!)(2!)]
=
840
So, # of "words" where the 4 I's are NOT together =
69,300 -
840 =
68,460
Cheers,
Brent
