Question

The number of groups that can be made from $5$ different green balls, $4$ different blue balls and $3$ different red balls, if at least $1$ green and $1$ blue ball is to be included?

• A. $3700$
• B. $3720$
• C. $4340$
• D. None of these

Answer 1

The correct option is B

$3720$

Explanation for the correct option:

Find the required number of groups.

Three groups of $5$ different green balls, $4$ different blue balls, and $3$ different red balls are given.

The total number of groups that can be formed out of given groups are $2^{12}$.

The total number of groups having no blue balls is $2^8$.

The total number of groups having no green balls is $2^7$.

The total number of groups having no green balls and no blue balls is $2^3$.

So, the total number of groups with at least $1$ green and $1$ blue balls is given by $2^{12}-(2^7+2^8-2^3)=3720$.

Therefore, The number of groups that can be made from $5$ different green balls, $4$ different blue balls, and $3$ different red balls, if at least $1$ green and $1$ blue balls are to be included is $3720$.

Hence, option $B$ is the correct option.