Permutations and Combination

Roland2rule wrote:What fraction of seven lettered words formed using the letters of the words CLASSIC will have the two C's always together?

A) 2/7
B) 5/7
C) 15/19
D) 4/19
E) 2/8

OA is A
Is there a unique formula to solve this question? An Expert answer is needed here.

First determine how many 7 letter "words" can be formed with these 7 letters.

The first letter can be any of the 7. The next any of the remaining 6 and so on. So 7x6x5x...1 = 7!

Recognize that two of the letters, C, are the same, so need to divide by 2.

So the total possible 7 letter words are 7!/2. That is the denominator of the fraction.

Now, determine how many of these have 2 C's together. Start with the two CCs in positions one and two. The remaining 5 letters can be arranged as discussed above 5x4x3x...1 = 5!

Now, march the two CC's through the letter positions and you can see there are 6 in total. So the total number of words with CC's together is 6x5! = 6!

Divided 6! by 7!/2 =6!*2/7! = [spoiler] 2/7, A[/spoiler]