There are 8 books in a shelf that consist of 2 paperback

Question

There are 8 books on a shelf, of which 2 are paperbacks and 6 are hardbacks. How many possible selections of 4 books from this shelf include at least one paperback?

A. 40
B. 45
C. 50
D. 55
E. 60

There are 8 books in a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random without replacement, how many different combinations are there that at least one paperback book is selected?

A. 40
B. 45
C. 50
D. 55
E. 60

Attachments

304B6C99-3734-4679-B856-74905CB3D357.jpeg [ 784.72 KiB | Viewed 26079 times ]

Bunuel · Accepted Answer

It's almost always better to solve "at least one" combination/probability questions using the formula {at least one} = {total} - {none}.

The total number of selections of 4 books out of 8:  C^4_8 = 70 ;
The total number of selections of 4 hardback books out of 6 (so none are paperback):  C^4_6 = 15 ;

Therefore, {at least one} = {total} - {none} = 70 - 15 = 55.

Answer: D.

srivas · Answer

There are 8 books in a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random without replacement, how many different combinations are there that at least one paperback book is selected?

A. 40
B. 45
C. 50
D. 55
E. 60

Attachments

304B6C99-3734-4679-B856-74905CB3D357.jpeg [ 784.72 KiB | Viewed 26079 times ]

Soln:
No. of combinations which have atleast one paperback is
= Total no. of combinations - Combinations in which there is no paperback
= 8C4 - 6C4
= 55 ways

Other way of doing this is finding out directly

No. of combinations which have atleast one paperback is
= Total no. of combinations that have 2 paperback + total no. of combinations that have 1 paperback
= 2C2 * 6C2 + 2C1 * 6C3
= 15 + 40
= 55 ways

sidbidus · Answer

No of combinations = p(1) + p(2)

= (2C1 * 6C3) + (2C2 * 6C2)

= (2*6*5*4)/(3*2*1) + (6*5)/2

= 40 + 15

= 55

beckee529 · Answer

i would go about it this way:

total - all hardback
total = 8C4
hardback = 6C4

8C4 - 6C4 
70-15 = 55

Rizz · Answer

The selection should at least contain one PAPERBACK book. There are a total of 2 PB and 6 HB books are available.

The combinations that at least one PB book will come out are:  PHHH & PPHH

1. PHHH = 2C1 * 6C3 = 2 * (5*4) = 40 
 2. PPHH = 2C2 * 6C2 = 1 * ((6*5) / 2) = 15

In total 40+15 = 55 Ways

manimgoindown · Answer

Hmm....so when I did this I made an error that might have carried over from using combinatorics from probability. I found the total as 8C4 and to find the amount to subtract from it I had (6C4)(2C0). Why don't we multiply by the books not chosen and what's would it mean if we did?

I'm trying to understand a fundamental flaw I made here

I'm trying to understand a fundamental flaw I made here

Bunuel · Answer

You'd get the same result because 2C0=2!/(2!*0!)=2/(2*1)=1: there is one way to choose 0 books from 2 books.

Hope it's clear.

mbaapp1234 · Answer

There are two scenarios that fit the question: either one paperback or two is selected.

If one is selected, then the total number of combinations is 2C1*6C3 + 2C2*6C2

2C1*6C3:
2*6!/(3!*3!) = 2*6*5*4/3*2 = 40

2C2*6C2
1*6*5*4*3*2/(2*4*3*2) = 15

40+15 =55

eishan123 · Answer

Could someone provide a group of similar practice problems? Would really appreciate the help on this. Thanks

Bunuel · Answer

21. Combinatorics/Counting Methods 
     Theory   
 Math: Combinatorics  
 Combinatorics Made Easy! 
 1 hour Combinatorics Video Lesson 
 Combinations while dealing with similar and dissimilar things 
 When Permutations & Combinations and Data Sufficiency Come Together 
     Questions   
 Advanced Topic: Probability and Combinations Questions With Solutions 
 Letter Arrangements in Words 
 Seating Arrangements in a Row and around a Table 
 Constructing Numbers, Codes and Passwords 
  DS Questions 
 PS Questions

22. Probability 
     Theory   
 Math: Probability 
 Probability Made Easy! 
 GMAT Tip of the Week: 99 Problems But Probability Ain't One 
     Questions   
 Probability and Geometry Problems 
 Advanced Topic: Probability and Combinations Questions With Solutions 
  DS Questions 
 PS Questions

For more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

JeffTargetTestPrep · Answer

We can use the equation:

The number of ways in which at least 1 paperback book is selected = The total number of ways to select 4 books – The number of ways in which no paperback books are selected
 
The number of ways in which no paperback books are selected is equivalent to the number of ways in which all 4 books selected are hardcover. Let’s determine that now. There are 6 hardback books, and 4 must be selected; thus:

6C4 = 6!/(4! x 2!) = (6 x 5)/2! = 30/2 = 15 ways

Now we determine the total number of ways to select the books. There are 8 total books and 4 must be selected, thus:
 
8C4 = 8!/(4! x 4!) = (8 x 7 x 6 x 5)/4! = 7 x 2 x 5 = 70 ways
 
Thus, the number of ways to select at least one paperback book = 70 – 15 = 55.
 
Answer: D

Rakhi08 · Answer

Hi Bunuel  Bunuel ,
Here why cant we do like below -
Total = 8*7*6*5
Only Hardback = 6*5*4*3

Required = Total - Hardback

Bunuel · Answer

Say we have A, B, C, D, E, F, G, and H books.

With 8*7*6*5 you'd get duplications in 4-book selections. If we choose A first, then choose B, then choose C and finally choose D, you'd get the same four books as if we choose D first, then choose A, then choose B, and then choose C. Note that the order of the books does not matter. Your method assumes that the order of the books does matter - you're choosing one book as the "first book", another as the "second book" and so on. Thus you get duplications. In order to get rid of them, you should divide 8*7*6*5 by 4!.

BrentGMATPrepNow · Answer

In general, when it comes to counting questions involving "at least," we can use the following property: 
Number of outcomes that satisfy the restriction = (number of outcomes that ignore the restriction) - (number of outcomes that BREAK the restriction)

So we get: 
 Number of combinations with at least one paperback = (total number of 4-book combinations) - (number of 4-book combinations that DON'T have at least one paperback)

In other words: 
 Number of combinations with at least one paperback = ( total number of 4-book combinations ) - ( number of 4-book combinations with ZERO paperbacks )

total number of 4-book combinations   
There are 8 altogether, and we must choose 4 of them
Since the order in which we select the books does not matter, we can use combinations. 
We can select 4 books from 8 books in 8C4 ways (=   70  )

number of 4-book combinations with ZERO paperbacks   
In order to get ZERO paperbacks, we'll select 4 books from the 6 hardback books only.
We can select 4 books from 6 books in 6C4 ways (=   15  )

So, the  number of combinations with at least one paperback = ( 70 ) - ( 15 )  =  55

Answer: D

MBAHOUSE · Answer

There are 8 books in a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random without replacement, how many different combinations are there that at least one paperback book is selected?

A. 40
B. 45
C. 50
D. 55
E. 60

Attachments

304B6C99-3734-4679-B856-74905CB3D357.jpeg [ 784.72 KiB | Viewed 26079 times ]

A. 40
B. 45
C. 50
D. 55
E. 60

DanTheGMATMan · Answer

Total minus choosing all hardbacks:

https://www.youtube.com/watch?v=X93QGX_M3Ac

Hrley · Answer

I'd like to introduce my way of doing this. It combines both the probability approach and the combinatorics approach.

Probability of selecting 4 books that at least 1 of them is paperback = 1 - Prob that none of them are paperback
This is 1 - 6/8 * 5/7 * 4/6 * 3/5 = 11/14

The total possible combinations of choosing 4 out of 8 books is 8C4 = 70

=> No. combinations of selecting 4 books that at least 1 of them is paperback = 70 * 11/14 = 55

There are 8 books in a shelf that consist of 2 paperback

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

There are 8 books in a shelf that consist of 2 paperback

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards