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# A “phrase” is made up of the first seven letters of the alphabet so

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Math Expert
Joined: 02 Sep 2009
Posts: 59590
A “phrase” is made up of the first seven letters of the alphabet so  [#permalink]

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21 Jul 2017, 05:04
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Difficulty:

95% (hard)

Question Stats:

42% (02:56) correct 58% (02:20) wrong based on 44 sessions

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A “phrase” is made up of the first seven letters of the alphabet so that each letter appears exactly once. The “phrase” must contain at least two “words”, which must contain at least two letters. For instance, ABC DE FG and AB CDE FG are distinct “phrases”. How many distinct “phrases” result given the conditions above?

(A) 5040

(B) 10,080

(C) 7!(5)

(D) 7!(6)

(E) 7!(7)

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Re: A “phrase” is made up of the first seven letters of the alphabet so  [#permalink]

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22 Jul 2017, 12:41
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Bunuel wrote:
A “phrase” is made up of the first seven letters of the alphabet so that each letter appears exactly once. The “phrase” must contain at least two “words”, which must contain at least two letters. For instance, ABC DE FG and AB CDE FG are distinct “phrases”. How many distinct “phrases” result given the conditions above?

(A) 5040

(B) 10,080

(C) 7!(5)

(D) 7!(6)

(E) 7!(7)

The “phrase” must contain at least two “words”, which must contain at least two letters.Thus , the possibilties are
3-2-2 -> ABC DE FG
2-3-2 -> AB CDE FG
2-2-3 -> AB CD EFG
3-4 -> ABC DEFG
4-3 -> ABCD EFG
5-2 -> ABCDE FG
2-5 -> AB CDEFG

In each of above scenarios, the letters can be arranged in 7 different ways.
Total number of ways = 7!*7

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Re: A “phrase” is made up of the first seven letters of the alphabet so  [#permalink]

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21 Jul 2017, 05:36
IMO answer is E. But I will wait for OA-

--First we need to know that for 7 letter word-7! is the number of words that can be formed. But now because you need to divide these 7! words into phrases.

-- There are 3 possible division possible - [3,2,2] eg. ABC DE FG , [5, 2] eg. ABCDE FG , [3,4] eg. ABC DEFG .

-- First case [3,2,2] - will result in 3 x 7! different phrases ( Note- it should not be 3! x 7!- I was making that mistake of permuting the 3 words. Why? see this example ABC DE FG & ABC FG DE are different phrases but they are already counted because of letter permutation)

-- Second case [ 5,2] - will result in 2 x 7! different phrase

-- third case [ 3,4] - will result in 2 X 7 ! different phrase
Re: A “phrase” is made up of the first seven letters of the alphabet so   [#permalink] 21 Jul 2017, 05:36
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