VJesus12 wrote:There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 Portuguese books. What is the probability of choosing 2 books in different languages?
A. 31/90
B. 3/10
C. 1/3
D. 31/45
E. 28/90
[spoiler]OA=D[/spoiler]
Source: Economist GMAT Tutor
We can use the formula:
P(choosing two books of different language) = 1 - P(choosing two books of same language)
So our options are 2 English, 2 Spanish or 2 Portuguese books.
2 English books can be chosen in:
5C2 = (5 x 4)/2! = 10 ways
2 Spanish books can be chose in:
3C2 = 3 ways
2 Portuguese books can be chosen in:
2C2 = 1 way
The total ways to select 2 books from 10 is:
10C2 = (10 x 9)/2 = 45 ways
So P(choosing two books of same language) = (3 + 10 + 1)/45 = 14/45.
Thus, P(choosing two books of different language) = 1 - 14/45 = 31/45.
Alternate Solution:
If the first book is an English book (for which there is a 5/10 = 1/2 probability), the second book can be any of the 3 + 2 = 5 books among the 9 remaining books. Therefore, the probability of choosing an English book followed by a book in a different language is 1/2 x 5/9 = 5/18.
If the first book is a Spanish book (for which there is a 3/10 probability), the second book can be any of the 5 + 2 = 7 books among the 9 remaining books. Therefore, the probability of choosing an Spanish book followed by a book in a different language is 3/10 x 7/9 = 21/90 = 7/30.
If the first book is a Portuguese book (for which there is a 2/10 = 1/5 probability), the second book can be any of the 5 + 3 = 8 books among the 9 remaining books. Therefore, the probability of choosing an Portuguese book followed by a book in a different language is 1/5 x 8/9 = 8/45.
Therefore, the total probability of choosing two books in different languages is 5/18 + 7/30 + 8/45 = 25/90 + 21/90 + 16/90 = 62/90 = 31/45.
Answer: D
