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In the Land of Oz only one or two-letter words are used. The

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In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132
[Reveal] Spoiler: OA

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Re: In the Land of Oz only one or two-letter words [#permalink]

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New post 02 Jun 2012, 03:42
I am getting the answer as 131 - D

Earlier there were 66 letters
No of two letter words = 66*65
Single letter word = 66
Total = 66^2

Now there were 65 letters
No of two letter words = 65*64
Single letter word = 65
Total = 65^2

Diff = 66^2 - 65^2 = 131 = lost words

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Alternate solution

The seventh letter (Let it be @) can be used with other 65 in two type of arrangents
one when seventh letter is in 1st slot @_ (65 ways)
another, when seventh letter is in 2nd slot _@ (65 ways)

seventh letter (@) itself

Lost words = 1 + 65 + 65 = 131

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Re: In the Land of Oz only one or two-letter words [#permalink]

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New post 02 Jun 2012, 04:19
Number of one letter words that can be made using 66 letters = 66
Number of two letter words that can be made using 66 letters = 66C2*2!
Total number of words using all 66 letters = 66 + 66C2 * 2!

Number of one letter words that can be made using 65 letters = 65
Number of two letter words that can be made using 65 letters = 65C2*2!
Total number of words using 65 letters = 65 + 65C2*2!

Number of words that the language has lost = 66 + 66C2*2! - (65+65C2*2!)
= 1 + 130
= 131

Choice D
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New post 02 Jun 2012, 04:24
GyanOne wrote:
Number of one letter words that can be made using 66 letters = 66
Number of two letter words that can be made using 66 letters = 66C2*2!
Total number of words using all 66 letters = 66 + 66C2 * 2!

Number of one letter words that can be made using 65 letters = 65
Number of two letter words that can be made using 65 letters = 65C2*2!
Total number of words using 65 letters = 65 + 65C2*2!

Number of words that the language has lost = 66 + 66C2*2! - (65+65C2*2!)
= 1 + 130
= 131

Choice D



official answer is Definitely E , I have rechecked . Let's wait for expert solution.

Mean while please tell me how you got the total number of 2 letter words , Permutation tells me (66!)/(64!) which gives me 66*65 . But I can't figure out 65C2*2!

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In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132

The answer to the question is indeed E. The problem with above solutions is that they do not consider words like AA, BB, ...

The number of 1 letter words (X) that can be made from 66 letters is 66;
The number of 2 letter words (XX) that can be made from 66 letters is 66*66, since each X can take 66 values.
Total: 66+66*66.

Similarly:
The number of 1 letter words (X) that can be made from 65 letters is 65;
The number of 2 letter words (XX) that can be made from 66 letters is 65*65, since each X can take 65 values.
Total: 65+65*65.

The difference is (66+66*66)-(65+65*65)=132.

Answer: E.

Hope it's clear.
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New post 02 Jun 2012, 07:18
Bunuel on target as always
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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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New post 02 Jun 2012, 07:25
Bunuel wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132

The answer to the question is indeed E. The problem with above solutions is that they do not consider words like AA, BB, ...

The number of 1 letter words (X) that can be made from 66 letters is 66;
The number of 2 letter words (XX) that can be made from 66 letters is 66*66, since each X can take 66 values.
Total: 66+66*66.

Similarly:
The number of 1 letter words (X) that can be made from 65 letters is 65;
The number of 2 letter words (XX) that can be made from 66 letters is 65*65, since each X can take 65 values.
Total: 65+65*65.

The difference is (66+66*66)-(65+65*65)=132.

Answer: E.

Hope it's clear.


+1 awesome , solution

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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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New post 02 Jun 2012, 19:24
Bunuel wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132

The answer to the question is indeed E. The problem with above solutions is that they do not consider words like AA, BB, ...

The number of 1 letter words (X) that can be made from 66 letters is 66;
The number of 2 letter words (XX) that can be made from 66 letters is 66*66, since each X can take 66 values.
Total: 66+66*66.

Similarly:
The number of 1 letter words (X) that can be made from 65 letters is 65;
The number of 2 letter words (XX) that can be made from 66 letters is 65*65, since each X can take 65 values.
Total: 65+65*65.

The difference is (66+66*66)-(65+65*65)=132.

Answer: E.

Hope it's clear.


silly mistakes as ever..... :oops:
Thanks Bunuel

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Let X be the seventh letter that will no longer be in use. Number of one letter words that use X =1. Number of two letter words that use X = 131 (65 with X as first letter, 65 w X as second letter and one as XX). This gives us a total of 1+131 =132 words that use X and which will no longer be available.

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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


Good Question.

An important thing to remember in such words/numbers/passwords questions is the step by step process.

Say we need to make 1 or 2 letter words, we need to consider 2 things if 'distinct letters' is not mentioned: the number of letters and repetitions allowed. We tend to ignore 'repetitions' because most questions say 'distinct letters'. The reason is that all different kinds of repetitions allowed need to be taken care of separately which makes the question cumbersome. GMAT doesn't give cumbersome questions. The good thing about this question is that there is only word which can have repetitions so calculations are not cumbersome but it is easy to ignore that word and arrive at the incorrect answer.

Say if G is discarded,

One letter words
Only 1 - G

Two letter words - all unique letters
65 different letters can be chosen alone with G and they can be arranged in 2! ways so 65*2 = 130

Two letter words - with repetition
Only 1 - GG

Total = 132
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Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


There are quite a few ways to solve this problem. Bunuel showed one, Karishma another. I solved this a slightly different way, so feel free to use whichever method makes the most sense to you on test day. There is probably nothing more important on the GMAT quant section than mental agility, i.e. being able to solve any question in a variety of ways (algebra, concept, picking numbers, back solving, etc)

Let's break it down into 1-letter codes and 2-letter codes. 1 letter codes is trivial. You're losing one letter (let's call it G since it's the 7th letter).

The real magic is the two letter codes. If you drop G, you're losing every single word that has G_ or _G. That means 66 letters for G in the first spot and 66 letters for G in the second spot. However you're double counting GG, so you have to remove one. Thus for 2-letter codes you lose (66+66-1) = 131.

Adding back in the 1 for 1-letter codes gives a total of 132. Very similar to a probability or Venn diagram question where you often double count elements.

Hope this helps!
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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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New post 10 Nov 2013, 13:23
hi, i solved it that way :

Total words lost = total words with 66 letters - total words with 65 letters

1) total words with 66 letters = 66 (total words with 1 letter) + 66P64 (total words with 2 different letters : AB, AC etc.) + 66 (total words of 2 identical letters : AA, BB etc.) = \(66+66*65+66\) = \(66*2+66*65\)

2) total words with 65 letters = 65 (total words with 1 letter) + 65P64 (total words with 2 different letters : AB, AC etc.) + 65 (total words of 2 identical letters : AA, BB etc.) = \(65+65*64+65\) = \(65*2+65*64\)

Finally the answer is :66*2+66*65 (1) -65*2-65*64 (2) = 2 + 65*(66-64) = 132

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New post 10 Jul 2016, 14:09
I'm late to the party but whatever...

Total number of one-letter words without restriction: 66
Total number of two-letter words without restriction (order matters): 66P2 = 66*65
Total number of one-letter and two-letter words without restriction: 66*65 + 66

Total number of one-letter words with restriction: 65
Total number of two-letter words with restriction (order matters): 65P2 = 65*64
Total number of one-letter and two-letter words with restriction: 65*64 +65

-> Total number of words lost: 65*(66-64) + 66 - 65 = 131 (D)

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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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New post 28 Nov 2016, 05:34
Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


The question must specify HOW ARE THE WORDS FROM THE LETTERS CONSTRUCTED. Whether repetition is allowed or not allowed must be specified. Bunuel please see.
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New post 28 Nov 2016, 05:48
ShashankDave wrote:
Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


The question must specify HOW ARE THE WORDS FROM THE LETTERS CONSTRUCTED. Whether repetition is allowed or not allowed must be specified. Bunuel please see.


If the repetition were not allowed it would have been mentioned.
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Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


We are given that in the Land of Oz only one- or two-letter words are used. If all 66 letters can be used, then we have:

1-letter words = 66

2-letter words = 66^2

Thus, there are 66 + 66^2 words if all 66 letters can be used.

When the seventh letter is taken away, we have:

1-letter words = 65

2-letter words = 65^2

Thus, there are 65 + 65^2 words when the seventh letter is taken away.

The number of lost words is:

(66 + 66^2) - (65 + 65^2)

66 + 66^2 - 65 - 65^2

1 + 66^2 - 65^2

1 + (66 - 65)(66 + 65)

1 + (1)(131) = 132

Answer: E
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Re: In the Land of Oz only one or two-letter words are used. The [#permalink]

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New post 18 Jun 2017, 23:36
Joy111 wrote:
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?

A. 65
B. 66
C. 67
D. 131
E. 132


Two ways to do this one.

1.
with 66 letters, how many one or two letter words can we make?

\(67 * 66\) -------(1)

we can fill the first position with 67 different things(66 letters or a blank), and the second position with 66 different letters.

With 65 letters after removing the 7th letter, how many words do we have?

\(66 * 65\) -------(2)

(1) - (2)

= 132(E)

2.
Now that we have removed the 7th letter, we have lost all the single and double letter words that we could make which include this 7th letter.

Single letter words with 7th letter?

\(1\)

Double letter words?

\(2 * 65 + 1\) ---Double letter words by using 7th letter only once + Double letter word by using 7th letter twice

Total = 132(E)
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