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Question

$$7$$ white balls and $$3$$ black balls are placed in a row at random. The probability that no two black balls are adjacent is


A
12
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B
715
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C
215
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D
13
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Solution

The correct option is B $$\dfrac{7}{15}$$
We will put black balls in the gaps to ensure no two block balls are adjacent 7 white balls
_W_W_W_W_W_W_W_
8 gaps
To choose 3 out of 8 gaps $$^8C_3$$ ways
Probability = $$\frac { 8\quad favourable }{ Total } $$
= $$\displaystyle \frac { ^{ 8 }{ C_{ 3 } } }{ _{  }^{ 10 }{ C_{ 3 } } } $$
= $$\displaystyle \frac { 8!3!7! }{ 3!5!10! } =\frac { 6\times 7 }{ 9\times 10 } =\frac { 7 }{ 15 } $$

Maths

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