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# There are 10 books on a shelf: 5 English books, 3 Spanish books and 2

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There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 12:15
1
3
00:00

Difficulty:

35% (medium)

Question Stats:

75% (01:39) correct 25% (01:57) wrong based on 193 sessions

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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

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There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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Updated on: 13 Jul 2015, 22:56
4
1
reto 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

Probability = Favourable outcome / Total Outcome = 1- (Unfavourable outcome / Total Outcome)

METHOD-1

Favourable Outcomes

Case-1: One book is English (out of 5 English books) and one book is Spanish (out of 3 Spanish books) i.e. 5C1*3C1
Case-2: One book is Spanish (out of 3 Spanish books) and one book is Portuguese (out of 2 Portuguese books) i.e. 3C1*2C1
Case-3: One book is English (out of 5 English books) and one book is Portuguese (out of 2 Portuguese books) i.e. 5C1*2C1

Total favourable Outcomes = 5C1*3C1+3C1*2C1+5C1*2C1 = 15+6+10 = 31
Total Possible Outcomes = 10C2 = 45
i.e. Probability = 31/45

METHOD-2(Taking arrangement of books into account

Case-1: One book is English (out of 5 English books) and one book is Spanish (out of 3 Spanish books) i.e. 5*3*2! (with arrangement)
Case-2: One book is Spanish (out of 3 Spanish books) and one book is Portuguese (out of 2 Portuguese books) i.e. 3*2*2! (with arrangement)
Case-3: One book is English (out of 5 English books) and one book is Portuguese (out of 2 Portuguese books) i.e. 5*2*2! (with arrangement)

Total favourable Outcomes = 5*3*2!+3*2*2!+5*2*2! = 30+12+20 = 62
Total Possible Outcomes = 10*9 = 90
i.e. Probability = 62/90 = 31/4

METHOD-3

Unfavourable Outcomes

Case-1: Both selected books are English books (out of 5 English books) = 5C2
Case-2: Both selected books are Spanish books (out of 3 English books) = 3C2
Case-3: Both selected books are Portuguese books (out of 2 Portuguese books) = 2C2

Total Unfavourable Outcomes = 5C2+3C2+2C2 = 10+3+1 = 14
Total Possible Outcomes = 10C2 = 45
i.e. Probability = 1- (14/45) = 31/45
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Originally posted by GMATinsight on 13 Jul 2015, 22:48.
Last edited by GMATinsight on 13 Jul 2015, 22:56, edited 1 time in total.
##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 51214
Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 12:21
2
1
reto 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

ES + EP + SP = 2*5/10*3/9 + 2*5/10*2/9 + 2*3/10*2/9 = 31/45. Each time we are multiplying by 2 because each case has two ways of occurring. For example, English-Spanish can be picked as English first then Spanish as well as Spanish first then English.

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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 12:43
Bunuel wrote:
reto 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

ES + EP + SP = 2*5/10*3/9 + 2*5/10*2/9 + 2*3/10*2/9 = 31/45. Each time we are multiplying by 2 because each case has two ways of occurring. For example, English-Spanish can be picked as English first then Spanish as well as Spanish first then English.

Bunuel,
You multiply by 2 each case has two ways of occurring. But Is important to pick English first then Spanish or Spanish first then English? I think both are the same so we do not need to multiply by 2.

I hope you can help.

Thanks
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Joined: 02 Sep 2009
Posts: 51214
Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 12:47
Mo2men wrote:
Bunuel wrote:
reto 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

ES + EP + SP = 2*5/10*3/9 + 2*5/10*2/9 + 2*3/10*2/9 = 31/45. Each time we are multiplying by 2 because each case has two ways of occurring. For example, English-Spanish can be picked as English first then Spanish as well as Spanish first then English.

Bunuel,
You multiply by 2 each case has two ways of occurring. But Is important to pick English first then Spanish or Spanish first then English? I think both are the same so we do not need to multiply by 2.

I hope you can help.

Thanks

The point is that English-Spanish CAN occur in two ways, so we should account for both cases. It's not a matter of importance.
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 13:00
The point is that English-Spanish CAN occur in two ways, so we should account for both cases. It's not a matter of importance.[/quote]

I used "important" by mistake. I mean is it a matter of arrangement? I still did not get your point. Can you please elaborate your point with an example ?

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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 21:31
1
reto 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

My attempt -

Ways of choosing two books from 3 groups(one book from English AND other from Spanish OR one book from Spanish AND other from Portuguese OR one book from English AND other from Portuguese ) = 5C1*3C1+3C1*2C1+5C1*2C1 = 15+6+10 = 31

Total ways of choosing from 10 books = 10C2 = 45

Probability of choosing 2 books in different languages = 31/45.
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There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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13 Jul 2015, 22:53
1
Mo2men wrote:
The point is that English-Spanish CAN occur in two ways, so we should account for both cases. It's not a matter of importance.

I used "important" by mistake. I mean is it a matter of arrangement? I still did not get your point. Can you please elaborate your point with an example ?

10C2 is 45 which is PURE SELECTION (WITHOUT ARRANGEMENT)

so We Don't need to take the arrangement of occurrence of two book English-Spanish for the given total outcomes (10C2) because total outcomes doesn't include arrangement

If you take two cases of spanish-English and English Spanish then you will have to take the total outcome in 10*9 = 90 ways
i.e. Total favourable Outcomes = 5*3*2!+3*2*2!+5*2*2! = 30+12+20 = 62
Total Possible Outcomes = 10*9 = 90
i.e. Probability = 62/90 = 31/45

I hope it helps!
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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26 Jul 2015, 02:39
balamoon:

Why didnt we multiply by 2 for each selection of book??
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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26 Jul 2015, 05:39
Shree9975 wrote:
balamoon:

Why didnt we multiply by 2 for each selection of book??

Why should we? Please explain. Keep in mind: order does not matter.
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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26 Jul 2015, 05:57
Shree9975 wrote:
balamoon:

Why didnt we multiply by 2 for each selection of book??

To add to what reto has mentioned above, selecting an English book and then a Spanish book will NOT be different from selecting a Spanish book and then an English book.

You only need to select books from 2 books .

Consider the question this way:

The desired probability = 1 - (probability of selecting 2 books of the same language) = $$1- \frac{5C2+3C2+2C2}{10C2}$$ = 31/45
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Posts: 7106
Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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26 Jul 2015, 07:10
Bunuel wrote:
reto 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

ES + EP + SP = 2*5/10*3/9 + 2*5/10*2/9 + 2*3/10*2/9 = 31/45. Each time we are multiplying by 2 because each case has two ways of occurring. For example, English-Spanish can be picked as English first then Spanish as well as Spanish first then English.

Hi,
isnt this Q slightly incomplete..
the wordings mean the prob of choosing two books of different languages..
I would take it to mean..
the 2 books of different languages can be chosen in following ways..
1) 3 books- one of one language and two of second language..
2) 4 books -2 of one language and two of second language..
3) 2 books- one of two languages each..
and so on

I think if the Q is supposed to mean what it actually wants to, the Q should be..
" what is the prob of the 2 books chosen to be of different languages?"
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There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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03 Aug 2015, 03:05
The probaility of taking two same books:

(5/10*4/9)+(3/10*2/9)+(2/10*1/9)=28/90

1-28/90=62/90=31/45

Combination approach:

(5C1*3C1)+(5C1*2C1)+(3C1*2C1)/10C2 = 15+10+6/45=31/45

D
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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03 Aug 2015, 05:40
1
Required probability = (5C1 * 3C1) + (5C1 * 2C1) + (3C1 * 2C1)/10C2 = 15 + 10 + 6/45 = 31/45. Ans (D).
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2  [#permalink]

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24 Nov 2018, 15:35
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Re: There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 &nbs [#permalink] 24 Nov 2018, 15:35
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