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Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2

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Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 02:12
1
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

62% (01:47) correct 38% (01:43) wrong based on 98 sessions

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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. 42800

Guys I've some concerns on this problem and I've put them under the spoiler. Please help me out!Thanks..

I am getting A. However, the OA is C. Below is the OE:

Explanation:

Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4) = (7C3 x 4C2)
={(7 x 6 x 5)x(4 x 3)}/ {(3 x 2 x 1)x(2 x 1)}
= 210.
Number of groups, each having 3 consonants and 2 vowels = 210.

Each group contains 5 letters.

Number of ways of arranging
5 letters among themselves = 5!
= 5 x 4 x 3 x 2 x 1
= 120.
Required number of ways = (210 x 120) = 25200.


I don't understand what is need of arranging the 5 letters again, as 7C3*4C2 will do that already. Can someone tell me where I'm going wrong?

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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 07:40
Using 4 x 3 / (2 x 1) = 6, to select your vowels... just calculated the number of groupings.

Example:
a e i o (4 vowels)
6 POSSIBLE GROUPINGS

a e
a i
a o
e i
e o
i o

If you take the ARRANGEMENT into account,
you should have 12 instead of 6.

a e
a i
a o
e i
e o
i o
e a
i a
o a
i e
o e
o i

But it's best to just get the possible number of grouping first which is 7C3 4C2 = 210. Then we arrange it by multiplying to 5!.. So as to allow consonants and vowels alternating...

Ex.

c d f e i

This will allow c d e i f.. Alternating elements...
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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 07:53
3
7C3 * 4C2 * 5!

(7 * 6 * 5)/3! * 4!/2!2! * 120



35 * 6 * 120

= 210 * 120

= 25200

I don't understand what is need of arranging the 5 letters again, as 7C3*4C2 will do that already.

7C3*4C2 will select letters, thereafter one has to arrange those.
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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 08:16
subhashghosh wrote:
7C3 * 4C2 * 5!

(7 * 6 * 5)/3! * 4!/2!2! * 120



35 * 6 * 120

= 210 * 120

= 25200

I don't understand what is need of arranging the 5 letters again, as 7C3*4C2 will do that already.

7C3*4C2 will select letters, thereafter one has to arrange those.


Uh.. thanks I think I got it now..

Say for example, the first combination is "r t y u i" You can sure arrange it in 5! ways, and since this is a unique combination, alphabets can be arranged in 5! and still form words not contained in any of the other combinations. Thanks for pointing out. great tip!
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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 17:44
7c3*4c2*5!

= 25200

Answer is C.
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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 04 May 2011, 19:32
5 letters in the word - hence 5! ways of arranging them.

positions - - - - -
7* 6 *5 * 4 * 3
consonants vowels

Thus without actually multiplying one can deduce like this : 5! = 120 means the number is divisible by 3.D and E POE.

Also, the number must have 2 zeros at the units and tens place, 120 and 6*5 = 30 is there.

C is a perfect fit.
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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2  [#permalink]

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New post 22 Apr 2018, 06:38
Top Contributor
gmatpapa 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. 42800


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 video on calculating combinations (like 4C2) in your head - see below

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 (= 25,200 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Re: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 &nbs [#permalink] 22 Apr 2018, 06:38
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