SOLUTION: If I pick 3 letters at a time from the word, MISSISSIPPI, how many combinations, I can get?

Question 666145: If I pick 3 letters at a time from the word, MISSISSIPPI, how many combinations, I can get?
Answer by swincher4391(1107) (Show Source): You can put this solution on YOUR website!
This is a multinomial problem. You can think of this in two ways. There is the intuitive way, and then there is the general formula.
In either case, how many letters are in MISSISSIPPI? 11.
Now partition the word into its letters.
1 M
4 Is
4 Ss
2 Ps
-----
11 total letters
Since the order in which our letters are partitioned doesn't matter, we can start with any letter.
Let's start with M.
We want to choose 1 M out of 11 letters. There are (11 choose 1) ways to do this. Now we have 10 letters to choose from. Let's use I now.
(10 choose 4) ways to choose Is from the remaining letters.
Continuing in this fashion we get:
(11 choose 1)(10 choose 4)(6 choose 4)(2 choose 2)= 34650.
If you were to break this apart for the general case, you'd eventually get to this result:
Let N be the number of total things to choose from. Let n1,n2,n3,n4...nk be the separate k partitions.
Then we have that the total number of ways to choose n1,n2,n3,n4....nk things from N is N!/(n1! * n2! * n3! ... nk!).
Then we get 11!/(4!*4!*2!*1!) = 34650, as we would expect.
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