Let the brain twisting commence!

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)

Give this challenging math problem a shot, and leave us a comment with you questions/concerns/compliments/explanations. Then, check back on Thursday for the answer and explanation. Good luck!

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Great job , I enjoy reading your articles Chris.

I could come up with the following cases:

1. 2 phrases

a) Length 2 and 5

b) Length 3 and 4

2. 3 phrases

a) Length 2, 2, 3

b) Length 1, 3, 3

3. 4 phrases

a) Length 1, 1, 2, 3

b) Length 1, 2, 2, 2

4. 5 phrases

a) Length 1, 1, 1, 2, 2

In all seven possible cases, so answer should be 7.(7!)

Guess it should be phrases with 2 words, 3 words etc.

Dear Anuj: This is Mike McGarry, Magoosh’s GMAT Expert. Chris will be back in a few days.

My friend, you got the right answer for the wrong reason, as you may appreciate. In the cases you list above, all the cases with a “one-letter word” are wrong because the problem specifies there must be at least two letters in a “word.” Thus, 2b is not allowed, and none of the 4 or 5 word phrases are allowed.

Furthermore, “words” in different orders are different phrases. Thus, ABCD EFG and EFG ABCD are two different phrases. Thus, the seven phrase possibilities are:

5-2

2-5

3-4

4-3

3-2-2

2-3-2

2-2-3

Into any of those seven groupings, we could permeate the seven letters and get unique phrases. Thus, (7!)*7 is the right answer for these reasons.

Does all this make sense?

Mike 🙂

I got 7!(7).

Conditions on the phrase structure yields 7 combinations and each combinations has 7! ways that the letters can be put together. So is it correct.

Dear Badhru,

This is Mike McGarry, the GMAT expert at Magoosh. Chris will be back in a few days. Yes, 7(7!) is the right answer, although there are right & wrong ways to get to that. See my answer to Anuj above. I am sure Chris will post correct answers soon.

Mike 🙂

Thank you, Mike.