swerve 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 OA is E
Source: Economist GMAT
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
Noting that the expression 66^2 - 65^2 is a difference of squares, we have:
1 + (66 - 65)(66 + 65)
1 + (1)(131) = 132
Alternate Solution:
Let x be the letter that is forbidden. Then, the people lost the following words: x, xx, ax, bx, cx, ... (there are 65 of them, one for all letters besides x) and xa, xb, xc, ... (there are 65 of them).
Thus, the people have lost 1 + 1 + 65 + 65 = 132 words.
Answer: E
