gmatt1476
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?
A. 96
B. 120
C. 144
D. 180
E. 240
We can use the following property:
# of outcomes that FOLLOW the rule = (# of outcomes that IGNORE the rule) - (# of outcomes that BREAK the rule)For this question, the property becomes:
# words with the C's separated = (# words using C, I, R, C, L, and E) - (# words with the C's adjacent)# words using C, I, R, C, L, and EIf we IGNORE the rule, then we're simply arranging the letters C, I, R, C, L, and E
--------------ASIDE--------------------------
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 (n = 11)
There are
4 identical I's (A = 4)
There are
4 identical S's (B = 4)
There are
2 identical P's (C = 2)
So, the total number of possible arrangements =
11!/[(
4!)(
4!)(
2!))]
--------------BACK TO THE QUESTION!--------------------------
In the word CIRCLE:
There are
6 letters in total (n = 6)
There are
2 identical C's (A = 2)
So, the total # words using C, I, R, C, L, and E = 6!/2! = 360# words with the C's adjacentTake the two C's and "glue" them together to create the single object CC. This ensures that the two C's are adjacent.
This means we must now arrange 5 unique objects (I, R, L, E, and CC).
We can arrange n different objects in n! ways
So, number of words with the C's adjacent = 5! = 120We're now ready for our calculation....
# words with the C's separated = (# words using C, I, R, C, L, and E) - (# words with the C's adjacent)= 360 - 120= 240Answer: E