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From the letters in MAGOOSH, we are going to make three-letter "words.

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Joined: 16 Dec 2018
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From the letters in MAGOOSH, we are going to make three-letter "words.  [#permalink]

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12 Nov 2019, 02:37
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55% (hard)

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From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720
Math Expert
Joined: 02 Aug 2009
Posts: 8309
From the letters in MAGOOSH, we are going to make three-letter "words.  [#permalink]

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12 Nov 2019, 07:12
1
hudacse6 wrote:
From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720

(I) Take all different letters.. M, A, G, O, S and H, so 6 letters...
They can be arranged in 6*5*4=120 ways
(II) two are O and third any of remaining 5..
So $$5*\frac{3!}{2!} =15$$ as each combination can be arranged in $$\frac{ 3!}{2!}$$ ways

Total 120+15=135

A
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Intern
Joined: 13 Jun 2019
Posts: 4
From the letters in MAGOOSH, we are going to make three-letter "words.  [#permalink]

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12 Nov 2019, 09:02
chetan2u wrote:
hudacse6 wrote:
From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720

(I) Take all different letters.. M, A, G, O, S and H, so 6 letters...
They can be arranged in 6*5*3=120 ways
(II) two are O and third any of remaining 5..
So $$5*\frac{3!}{2!} =15$$ as each combination can be arranged in $$\frac{ 3!}{2!}$$ ways

Total 120+15=135

A

I am confused a bit. why we can't consider 7 letters from the beginning and arrange them? I mean 7*6*5 = 210?
Math Expert
Joined: 02 Aug 2009
Posts: 8309
Re: From the letters in MAGOOSH, we are going to make three-letter "words.  [#permalink]

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12 Nov 2019, 09:18
1
merimanukyan wrote:
chetan2u wrote:
hudacse6 wrote:
From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720

(I) Take all different letters.. M, A, G, O, S and H, so 6 letters...
They can be arranged in 6*5*3=120 ways
(II) two are O and third any of remaining 5..
So $$5*\frac{3!}{2!} =15$$ as each combination can be arranged in $$\frac{ 3!}{2!}$$ ways

Total 120+15=135

A

I am confused a bit. why we can't consider 7 letters from the beginning and arrange them? I mean 7*6*5 = 210?

If you take 7 letters it will consist of both O.
Say the letters are M, A,G,O1,O2,S,H
So these 7*6*5 will consist of O1MA and O2MA. But both are same, OMA, so there are repetition in 7*6*5.
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Joined: 13 Jun 2019
Posts: 4
Re: From the letters in MAGOOSH, we are going to make three-letter "words.  [#permalink]

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12 Nov 2019, 09:20
Thank you! That makes sense!

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Re: From the letters in MAGOOSH, we are going to make three-letter "words.   [#permalink] 12 Nov 2019, 09:20
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