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# A reading list for a certain course consists of 10 books, of which 4 a

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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13 May 2015, 03:53
00:00

Difficulty:

55% (hard)

Question Stats:

68% (02:22) correct 32% (02:34) wrong based on 177 sessions

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A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195

Kudos for a correct solution.

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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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13 May 2015, 05:00
2
1
Can be selected as

Case 1 PPNN 2 Poems out of 4 and 2 Novels out of 6 i.e. 4C2 * 6C2 = 90
or
Case 2 PPPN 3 Poems out of 4 and 1 Novels out of 6 i.e 4C3 * 6C1 = 24
or
Case 3 PPPP 4 Poems out of 4 i.e. 4C4 =1

Total no. of selections case 1 + case 2+ case 3 = 90 + 24 +1 = 115

So answer must be B. 115
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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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13 May 2015, 07:25
3
1
Possibilities include
1novel, 3 poems
6! / 1!5! × 4! /3! 1! =24

2 novels, 3 poems
6!/2! 4! × 4!/2! 2! =90

0 novels, 4 poems
=1

24+90+1=115

B. 115
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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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13 May 2015, 11:18
1
Novel=N,Poem book=P; therefore 10=4+Novels =>N=6;P=4

The final selection would be = 2N and 2P +1N and 3P + 4P =6C2*4C2 + 6C1*4C3 + 4C4 = 90+24+1=115

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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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14 May 2015, 06:12
1
Bunuel wrote:
A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195

Kudos for a correct solution.

Options with two novels: 6C2 * 4C2 = 90

Options with one novel: 4C3 * 6C1 = 24

Option with no novel: 4C4 = 1

so 90+24+1 = 115

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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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15 May 2015, 02:23
Bunuel wrote:
A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195
[/i]

Ans: B

Solution: Selection requires at most 2 novels, means=
0N & 4P + 1N & 3P + 2N & 2P
6c0*4c4+ 6c1*4c3 + 6c2*4c2
= 1+ 24 + 90
= 115

Ans: B
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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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15 May 2015, 09:49
1
Bunuel wrote:
A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195

Kudos for a correct solution.

Required selection=(4C2*6C2)+(6C1*4C3)+(4C4)
=90+24+1
=115
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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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15 May 2015, 17:29

including at most 2 novels means that no novels included or one novels included or 2 novels included

no novels included =4C4= 1

one novels included = 4C3*6C1=24

2 novels included = 4C2*6C2=90

selections of 4 books satisfy the requirements= 1+24+90=115
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Posts: 58445
A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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18 May 2015, 07:41
1
1
Bunuel wrote:
A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195

Kudos for a correct solution.

OFFICIAL SOLUTION:

Total - restriction =

= (4 books out of 10) - ((3 novels and 1 poem) + (4 novels and 0 poems)) =

$$= C^4_{10} - (C^3_6*C^1_4 + C^4_6) =$$

= 210 - (20*4 + 15) = 115.

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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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31 May 2017, 19:02
Bunuel wrote:
A reading list for a certain course consists of 10 books, of which 4 are poems and the rest are novels. Each student is required to read a selection of 4 books from the list, including at most 2 novels. How many selections of 4 books satisfy the requirements?

A. 90
B. 115
C. 130
D. 144
E. 195

We are given that a course reading list contains 4 poems and 6 novels. We need to determine the number of ways a student can select 4 books if at most 2 novels can be selected.

Scenario one: 4 poems are selected.

4 poems can be selected in 4C4 = 1 way.

Scenario two: 1 novel and 3 poems are selected.

3 poems can be selected in 4C3 = 4!/[3!(4-3)!] = (4 x 3 x 2)/3! = 4 ways, and 1 novel can be selected in 6C1 = 6 ways.

Thus, 3 poems and 1 novel can be selected in 4 x 6 = 24 ways.

Scenario three: 2 novels and 2 poems are selected.

2 poems can be selected in 4C2 = 4!/[2!(4-2)!] = (4 x 3)/2! = 6 ways, and 2 novels can be selected in 6C2 = 6!/[2!(6-2)!] = (6 x 5)/2! = 15 ways.

Thus, 2 poems and 2 novels can be selected in 6 x 15 = 90 ways.

All 4 books with at most 2 novels can be selected in 1 + 24 + 90 = 115 ways.

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Re: A reading list for a certain course consists of 10 books, of which 4 a  [#permalink]

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13 Sep 2019, 02:57
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Re: A reading list for a certain course consists of 10 books, of which 4 a   [#permalink] 13 Sep 2019, 02:57
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