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.
Permutations and Combination
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First determine how many 7 letter "words" can be formed with these 7 letters.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.
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]
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The total number of seven-letter words that can be formed using the letters in CLASSIC (regardless if the 2C's are together or not) is calculated by using the indistinguishable permutations formula, noting that C appears twice and S appears twice in the word CLASSIC:BTGmoderatorRO 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.
7!/(2! x 2!) = 5040/(2 x 2) = 1,260.
The total number of seven-letter words that can be formed if the 2 C's are together (by considering the 2C's as one letter) is:
6!/2! = 720/2 = 360
Therefore, the fraction is 360/1,260 = 36/126 = 6/21 = 2/7.
Answer: A
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