M31-12

Question

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

Show Answer

Official Answer

B

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

kasturi72 · Accepted Answer

We can take 3 conditions
0 novels
1 novel 
2 novels
1st condition- 4c4- all poems = 1
2nd condition - 6c1 * 4c3 - 1 novel 3 poems = 6*4 = 24
3rd condition - 6c2 * 4c2 - 2 novels 2 poems = 15*6= 90 
Total = 115
So B

Bunuel · Answer

Official Solution:

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

Show Answer

Official Answer

B

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

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 .

Answer: B

hideyoshi · Answer

Bunuel , can you elaborate on how you came up with the restrictions, quoted & highlighted below,

Thanks.

Bunuel · Answer

Each student is required to read a selection of 4 books from the list, including  at most 2 novels . Therefore the restriction would be 3 novels and 1 poem AND 4 novels and 0 poems (basically all cases with more than 2 novels.).

Hope it's clear.

WillGetIt · Answer

Hello Bunuel,

Would you like to elaborate approach suggested by Kasturi72?

Thanks for your help.

manlog · Answer

These two approaches are equal. Bunuel's approach is to calculate 1) total number of combinations 2) combinations that doesn't satisfy the condition. Then subtract 2) from 1).

Kasturi's approach is to calculate all allowed combinations. Namely 0 novels, 1 novel, 2 novels.

HWPO · Answer

Hi Bunuel , maybe I am missing something, but would you explain why didn't you include a situation in which 2 poems are selected? **I am referring to the highlighted part in the quote. Thank you

Bunuel · Answer

In the solution we are calculating 1. the total number of selections of 4 books from 10 ( C^4_{10} ) and 2. the number of selections ( C^3_6*C^1_4 + C^4_6 ) that does NOT satisfy the condition (at most 2 novels). Then we subtract 2 from 1.

The direct way would be to 0 novels and 4 poems, 1 novel and 3 poems and 2 novels and 2 poems and sum that.

DixitNiket · Answer

6C0*4C4 +6C1*4C3 + 6C2*4C2 =115

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

navyachopra · Answer

I don't agree with the explanation. what about the possibility of 2 novels and 2 poems

Bunuel · Answer

You are missing the point! The explanation already accounts for the possibility of selecting 2 novels and 2 poems. It calculates the total number of selections and subtracts the invalid combinations where more than 2 novels are selected. That's why only selections with 0, 1, or 2 novels are allowed, which includes the case you're asking about (2 novels and 2 poems).

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M31-12

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