Permutation-Combination
Aptitude

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Q.

A box contains $10$ balls out of which $3$ are red and rest are blue.

In how many ways can a random sample of $6$ balls be drawn from the bag so that at the most $2$ red balls are included in the sample and no sample has all the $6$ balls of the same colour?

 A.

$105$

 B.

$168$

 C.

$189$

 D.

$120$

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Solution:
Option(B) is correct

Six balls can be selected in the following ways:

One red ball and 5 blue ball

Or
Two red balls and 4 blue balls

Total number of ways:
$= ({^3C_1} × {^7C_5}) + ({^3C_2} × {^7C_4})$

$= 63 + 105$

$= \textbf{168}$


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