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
how many words
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- Jim@StratusPrep
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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
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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Take the task of creating 5-letter words and break it into stages.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
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:
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- 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
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- 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
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- https://www.beatthegmat.com/laniera-s-co ... 15764.html
Cheers,
Brent
- Jim@StratusPrep
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Oops, was a bit sleepy this morning. Thanks to Brent for the next step.
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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
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:Take the task of creating 5-letter words and break it into stages.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
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
GMAT/MBA Expert
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Good point. The question could be written to be less ambiguous.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
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
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Great question: I suppose because the answer we'd get if we DIDN'T make that assumption isn't one of the options.osama_salah wrote:Why did we assume that a letter can not be repeated?
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.