Each of the [6] squares above is to have exactly one letter and nothing else placed inside it. If 3 of the letters are to be the letter X, 2 of the letters are to be the letter Y and 1 of the letters is to be the letter Z, in how many different arrangements can the squares have letters placed in them?
(A) 30
(B) 60
(C) 108
(D) 120
(E) 720
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!)....]
So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
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!)]
---------------------------------------
Likewise, for this question, we can calculate the number of arrangements of the letters in XXXYYZ as follows:
There are
6 letters in total
There are
3 identical X's
There are
2 identical Y's
So, the total number of possible arrangements =
6!/[(
3!)(
2!)]
=
60
=
B
Here are two related questions:
-
https://www.beatthegmat.com/p-c-t202402.html
-
https://www.beatthegmat.com/permutation- ... 81906.html
Cheers,
Brent
