A box contains two white balls, three black balls and four red balls.

Question

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?

A. 16
B. 32
C. 64
D. 96
E. 128

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TarunKumar1234 · Accepted Answer

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?

A. 16
B. 32
C. 64
D. 96
E. 128

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Show Answer

Official Answer

C

rocky620 · Answer

Case-1   - 1 Black Ball & 2 Other balls
3c1*6c2 = 45

Case-2   - 2 Black Balls & 1 Other ball
3c2*6c1 = 18

Case-3   - 3 Black Balls
3c3 = 1

Total Cases = 45+18+1 = 64

IMO Option C

ScottTargetTestPrep · Answer

Solution:
 
Without any restrictions, the number of ways to choose 3 balls from 9 is 9C3 = (9 x 8 x 7)/(3 x 2) = 3 x 4 x 7 = 84.

Let’s calculate the number of ways we don’t select a black ball. Since there are 6 non-black balls, we can choose any 3 of them in 6C3 = (6 x 5 x 4) / (3 x 2) = 120/6 = 20 ways.

In summary, there are 84 different selections of 3 balls (if there are no restrictions), and 20 of them consist of selecting no black balls. Therefore, there must be 84 - 20 = 64 different selection of 3 balls with at least one black ball.

Answer: C

CrackverbalGMAT · Answer

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?

Total 9 balls are there ( 2W, 3B,4R)

Total no of ways to select 3 balls out of 9 = 9C3 = 9*8*7/3*2*1 = 84

Total no of ways to select 3 balls ,where there is no black ball selected i.e you should select from either white or red balls ( 2W + 4R) = 6C3= 20

Total no of ways to select 3 balls if atleast 1 black ball is included = Total no of ways to select 3 balls out of 9 - Total no of ways to select 3 balls ,where there is no black ball selected = 84 -20 = 64

Option C is the answer.

Thanks,
Clifin J Francis.

Paras96 · Answer

Total no. ways to select atleast 1 black ball = total ways of selection of any 3 balls from 9 balls without any restriction - total ways of selection 3 balls from group of only red and white balls
=9C3-6C3
=84-20
=64

Rohit_842 · Answer

What goes wrong when I say lets keep one black ball aside as we need "at least one black ball" and then select 2 balls out of 8. How am I counting incorrectly

Krunaal · Answer

You under estimate the count of black balls that way, first you could've picked 1 or 2 black balls from 3 black balls, now since you have "fixed" one black ball, you only have 2 black balls to pick from.

Case  
  Correct Count  
  Incorrect Count (Underestimation)  
 
  1 black, 2 non-black 
 3C1*6C2 = 45 
 6C2 = 15 
 
  2 black, 1 non-black 
 3C2*6C1 = 18 
 2C1*6C1 = 12 
 
  3 black 
 3C3 = 1 
 2C2 = 1 
 
   Total  
   64   
   28

sarthak1701 · Answer

Total Balls = 9 (WWBBBRRRR)

Ways to choose ANY 3 balls from 9 (without defining the color) = 9c3 = 84

Ways to NOT Pick any black ball = 6c3 = 20

Ways to choose at least one black ball = 84 - 20 = 64.

Ans C

PrecisionPrep · Answer

Is the answer C (64) based on the assumption that the balls of the same color are distinct/differentiated?

Bunuel · Answer

We don’t care which black or white balls are in the selection as long as at least one black ball is included in the draw. However, the balls are still physically distinct objects, so when calculating the number of ways to select, say, 3 balls out of 9, we still do 9C3. This is because when reaching for the balls in the box, there are still 9 distinct objects, so this gives us the number of ways to pick 3 distinct objects from 9.

A box contains two white balls, three black balls and four red balls.

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Case	Correct Count	Incorrect Count (Underestimation)
1 black, 2 non-black	3C1*6C2 = 45	6C2 = 15
2 black, 1 non-black	3C2*6C1 = 18	2C1*6C1 = 12
3 black	3C3 = 1	2C2 = 1
Total	64	28

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A box contains two white balls, three black balls and four red balls.

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