Hi All,

We're told the letters C, I, R, C, L, and E are to be used to form 6-letter strings such as CIRCLE or CCIRLE. We're asked for the number of different 6-letter strings that can be formed in which the two occurrences of the letter C are separated by AT LEAST one other letter. There are a couple of different ways to approach this question - and if you don't know an elegant way to approach it, then you can still get the correct answer with a little permutation math and some 'brute force.'

If the first letter is a C, then the second letter CANNOT be an C (that second letter would have to be one of the other 4 non-C letters)...

C 4

From here, any of the four remaining letters can be in the 3rd spot. After placing one, any of the remaining three letters can be in the 4th spot, etc. and the last letter would be in the 6th spot...

C 4 4 3 2 1

This would give us (4)(4)(3)(2)(1) = 96 possible arrangements with a C in the 1st spot.

If a non-C is in the 1st spot and a C is in the 2nd spot, then we have...

4 C _ _ _ _

A non-C would have to be in the 3rd spot (3 options), then any of the remaining three letters could be 4th, etc...

4 C 3 3 2 1

This would give us (4)(3)(3)(2)(1) = 72 possible arrangements

Next, we could have two non-Cs to start off, then Cs in following spots...

4 3 C 2 1 C --> (4)(3)(2)(1) = 24 possible arrangements
4 3 C 2 C 1 --> (4)(3)(2)(1) = 24 possible arrangements
4 3 2 C 1 C --> (4)(3)(2)(1) = 24 possible arrangements

This would give us an additional (3)(24) =72 possible arrangements

There are no other options to account for, so we have 96+72+72 = 240 total arrangements.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

BTGmoderatorDC wrote: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

OA E

Source: Official Guide

We can use the rule that says: TOTAL number of outcomes if we IGNORE the rule = (number of outcomes that FOLLOW the rule) + (number of outcomes that BREAK the rule)
In other words: Number of ways to arrange the 6 letter if we IGNORE the rule = (number of words that DON'T have adjacent C's) + (number of words that DO have adjacent C's)

Rearrange to get: number of words that DON'T have adjacent C's = (Number of ways to arrange the 6 letter if we IGNORE the rule) - (number of words that DO have adjacent C's)

Number of ways to arrange the 6 letter if we IGNORE the rule
If we IGNORE the rule, then we are arranging the letters in CIRCLE
Since we have DUPLICATE letters, we can apply the MISSISSIPPI rule (see video below)

In the word CIRCLE:
There are 6 letters in total
There are 2 identical C's
So, the total number of possible arrangements = 6!/(2!) = 360

number of words that DO have adjacent C's
Take the two C's and "glue" them together to get the SUPER LETTER "CC"
This ensures that the C's are together
We now must arrange CC, I, R, L, E
We can arrange n different objects in n! ways
So, we can arrange CC, I, R, L, and E in 5! ways (= 120 ways)
So, number of words that DO have adjacent C's = 120

So, number of words that DON'T have adjacent C's = 360 - 120 = 240

Answer: E

Cheers ,
Brent