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

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Re: A box contains two white balls, three black balls and four red balls. [#permalink]
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Bunuel wrote:
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

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.

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Re: A box contains two white balls, three black balls and four red balls. [#permalink]
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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

Thanks,
Clifin J Francis.
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Re: A box contains two white balls, three black balls and four red balls. [#permalink]
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
Re: A box contains two white balls, three black balls and four red balls. [#permalink]
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