how many words

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nahid078: Master | Next Rank: 500 Posts; Posts: 105; Joined: Tue May 13, 2014 10:41 amFollowed by: 1 member; Thanked: 2 times

by nahid078 » Mon Sep 08, 2014 4:22 am

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A. 210
B. 1050
C. 25200
D. 21400
D. None of these

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Jim@StratusPrep: MBA Admissions Consultant; Posts: 2279; Joined: Fri Nov 11, 2011 7:51 amFollowed by: 266 members; Location: New York; Thanked: 660 times; GMAT Score:770

by Jim@StratusPrep » Mon Sep 08, 2014 4:57 am

This is 7 choose 3 multiplied by 4 choose 2.

n!/(n-k)!k!

7 choose 3 = 7!/4!3! = 35

4 choose 2 = 4!/2!2! = 6

35 x 6 = 210

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Sep 08, 2014 5:22 am

nahid078 wrote:Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A. 210
B. 1050
C. 25200
D. 21400
E. None of these

Take the task of creating 5-letter words and break it into stages.

Stage 1: Select the 3 consonants to work with
Since the order in which we select the consonants does not matter, we can use combinations.
We can select 3 consonants from 7 consonants in 7C3 ways (= 35 ways)

Stage 2: Select the 2 vowels to work with
Since the order in which we select the vowels does not matter, we can use combinations.
We can select 2 vowels from 4 vowels in 4C2 ways (= 6 ways)

If anyone is interested, we have a free video on calculating combinations (like 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Stage 3: Take the 5 selected letters and arrange them.
We can complete this stage in 5! ways (= 120 ways).

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create all 5-letter words) in (35)(6)(120) ways ([spoiler]= 25200 ways[/spoiler])

Answer: C
--------------------------

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Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Jim@StratusPrep: MBA Admissions Consultant; Posts: 2279; Joined: Fri Nov 11, 2011 7:51 amFollowed by: 266 members; Location: New York; Thanked: 660 times; GMAT Score:770

by Jim@StratusPrep » Mon Sep 08, 2014 5:59 am

Oops, was a bit sleepy this morning. Thanks to Brent for the next step.

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osama_salah: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Wed Jun 15, 2016 2:05 am

by osama_salah » Mon Jul 04, 2016 7:56 am

Why did we assume that a letter can not be repeated?
e.g. out of B,C,D,F,G,H,J and A,E,I,O we can get:
BACAD and ECCEF

Brent@GMATPrepNow wrote:
nahid078 wrote:Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A. 210
B. 1050
C. 25200
D. 21400
E. None of these
Take the task of creating 5-letter words and break it into stages.

Stage 1: Select the 3 consonants to work with
Since the order in which we select the consonants does not matter, we can use combinations.
We can select 3 consonants from 7 consonants in 7C3 ways (= 35 ways)

Stage 2: Select the 2 vowels to work with
Since the order in which we select the vowels does not matter, we can use combinations.
We can select 2 vowels from 4 vowels in 4C2 ways (= 6 ways)

If anyone is interested, we have a free video on calculating combinations (like 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Stage 3: Take the 5 selected letters and arrange them.
We can complete this stage in 5! ways (= 120 ways).

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create all 5-letter words) in (35)(6)(120) ways ([spoiler]= 25200 ways[/spoiler])

Answer: C
--------------------------

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html

MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html

DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Jul 04, 2016 9:17 am

osama_salah wrote:Why did we assume that a letter can not be repeated?
e.g. out of B,C,D,F,G,H,J and A,E,I,O we can get:
BACAD and ECCEF

Good point. The question could be written to be less ambiguous.
In fact, we don't even know that the original 7 consonants and 4 vowels are all different.
The 11 letters could just be B,B,B,B,B,B,B,A,A,A,A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Thu Jul 07, 2016 3:50 pm

osama_salah wrote:Why did we assume that a letter can not be repeated?

Great question: I suppose because the answer we'd get if we DIDN'T make that assumption isn't one of the options.

That said, (E) would fit, so it's a poorly written Q. I'm guessing it's from the CAT (a rival to the GMAT), on which I've seen other poorly written Qs, but I can't say for sure.

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