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# D01-39

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Math Expert
Joined: 02 Sep 2009
Posts: 53734

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16 Sep 2014, 00:13
2
12
00:00

Difficulty:

45% (medium)

Question Stats:

66% (02:07) correct 34% (02:24) wrong based on 228 sessions

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Romi has a collection of 10 distinct books out of which 8 are small and 2 are large. In how many ways can he select 5 books to take with him on a trip if he can take at most 1 large book?

A. 56
B. 126
C. 152
D. 196
E. 252

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Math Expert
Joined: 02 Sep 2009
Posts: 53734

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16 Sep 2014, 00:13
3
2
Official Solution:

Romi has a collection of 10 distinct books out of which 8 are small and 2 are large. In how many ways can he select 5 books to take with him on a trip if he can take at most 1 large book?

A. 56
B. 126
C. 152
D. 196
E. 252

Since Romi can take at most 1 large book then he can take with him either 5 small books or 4 small books and 1 large book: $$C^5_8+C^4_8*C^1_2=56+140=196$$.

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Joined: 13 Nov 2011
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14 Dec 2015, 07:07
"C^5_8 = 56" how is this derived?
Math Expert
Joined: 02 Sep 2009
Posts: 53734

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17 Dec 2015, 09:23
LostinNY wrote:
"C^5_8 = 56" how is this derived?

Choosing 5 small books out of 8. So, Romi will have 5 small books and 0 large books.
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Joined: 08 Nov 2015
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17 Dec 2015, 19:08
Can you explain what the C^5/8 means. How do I get to 56 from this?
Math Expert
Joined: 02 Sep 2009
Posts: 53734

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20 Dec 2015, 11:35
dau245 wrote:
Can you explain what the C^5/8 means. How do I get to 56 from this?

C stands for combinations. Check for more here: math-combinatorics-87345.html
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Joined: 23 Apr 2016
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09 Jul 2016, 12:24
2
See example in the ebook from Magoosh team under Combinations: http://magoosh.resources.s3.amazonaws.c ... _eBook.pdf

C 5 over 8 = 8!/5!(8-5)! = 8*7*6*5*4*3*2*1/(5*4*3*2*1)*(3*2*1)
Simplify and you get 8*7*6/3*2*1
Simplify more and you get 4*7*2 = 8*7 = 56

You do the same with the other part of the equation.
So this for the first part. Hope this helps everyone!
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Joined: 09 May 2016
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15 Oct 2016, 11:00
1
Alternative solution
Since Romi can take at most 1 large book then he can take with him : All possible ways - (2 Large books and 3 books from 5 possible) =(5 small books from 10) - (3 small books from 8)=C510-C38=196
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Joined: 08 Mar 2016
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06 Mar 2017, 21:23
1
I have a question, for the second combination C^4_8, I get 70 not 140.
C^4_8 = 8*7*6*5 / 4*3*2*1 = 2*7*5 = 70
Math Expert
Joined: 02 Sep 2009
Posts: 53734

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07 Mar 2017, 02:30
jdlopez0224 wrote:
I have a question, for the second combination C^4_8, I get 70 not 140.
C^4_8 = 8*7*6*5 / 4*3*2*1 = 2*7*5 = 70

Yes, 8C4 = 70, but there we have 8C4 * 2C1 = 70*2 = 140.
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Joined: 06 Aug 2017
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11 Aug 2017, 12:17
I think this is a poor-quality question and I don't agree with the explanation. The answer seems to be wrongly calculated.
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Joined: 25 Feb 2013
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Location: India
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11 Aug 2017, 12:51
1
Victor__ wrote:
I think this is a poor-quality question and I don't agree with the explanation. The answer seems to be wrongly calculated.

Hi Victor__

it would be great if you could explain what exactly is wrong with the official solution?

As per the question there are two possibilities -

A. take $$1$$ large book $$AND$$ $$4$$ small books. Mathematically this can be represented as -
$$1$$ large book can be selected from $$2$$ large books in $$2_C_1$$ and remaining $$4$$ small books can be selected from $$8$$ small books in $$8_C_4$$
therefore total number of ways = $$2_C_1$$ $$*$$ $$8_C_4 = 140$$

$$OR$$

B. take $$5$$ small books and no large books. Mathematically it can be represented as -
$$5$$ small books can be selected from $$8$$ books in $$8_C_5 = 56$$

Therefore total number of ways = $$140+56 = 196$$
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Joined: 17 Sep 2013
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19 Jun 2018, 22:48
Hi there, i am relatively new to this group and could you please help me with the basics of this topic (may be with some link), i find this really difficult.
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regards,

honcho

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19 Jun 2018, 23:21
1
1
Intern
Joined: 22 Nov 2018
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31 Jan 2019, 10:16
2
Hi Bunuel , Could you please explain to me why is combination used here? Since the books are distinct shouldn't we use permutation i.e. 8*7*6*5*6 (4 small books and 2 large books for the last slot)
Manager
Joined: 12 Apr 2017
Posts: 64
Location: United States
Concentration: Finance, Operations
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07 Feb 2019, 12:44
This is a great question. I did not think about the possibility of 5 small books. Bookmarked for future practice.
Re: D01-39   [#permalink] 07 Feb 2019, 12:44
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# D01-39

Moderators: chetan2u, Bunuel

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