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Magoosh Brain Twister: Going Through a Phrase – Explanation

brain_twister

Welcome back! If you haven’t had a chance to try your hand at this week’s Brain Twister, head back to the original post so you don’t accidentally see the answer here.
 

Question

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)
 

Answer and Explanation:

Part of the challenge here is decoding what the problem is saying. Notice that both “phrase” and “words” are in quotation marks, meaning that combinations of the letters, e.g., ABC, aren’t actually words but just strings of letters, and when there is a space in between, e.g. BC DE

The best way to find the different ways in which we can split up “words” into “phrases” is to quickly list them out:

ABCDE FG

ABCD EFG

ABC DEFG

ABC DE FG

 

AB CDEFG

AB CD EFG

AB CDE FG

For each of these seven instances, the letters can be arranged in seven different ways. For instance, if we take the first case (ABCDE FG) so that the first five letters are part of one “word” and the last two letters form the second “word”, then there are 7! different ways we can arrange these letters. Below are two examples of such a configuration. Notice that the structure of the phrase (a five letter “word” is followed by a two letter “word”) stays the same.

BCADE  FG

GFEDC  BA

The 7!  ways applies to all 7 of the cases listed above, giving us 7(7!). Answer (E).

 

Thanks for playing. 🙂 See you in a couple weeks for a new challenge!

 

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