JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

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?

Choose the correct answer:

Explanation

1. Analysis by Color

Green Balls (5 different):

Each ball has 2 choices (In or Out). Total combinations = $2^5$ .

However, the condition states at least one green ball must be included. We subtract the case where no green ball is chosen (1 way).

$\text{Ways to choose green} = 2^5 - 1 = 31$
Blue Balls (4 different):

Similarly, each ball has 2 choices. Total combinations = $2^4$ .

The condition states at least one blue ball must be included. We subtract the case where no blue ball is chosen (1 way).

$\text{Ways to choose blue} = 2^4 - 1 = 15$
Red Balls (3 different):

There is no restriction on red balls. They can be included or not included in any number (0, 1, 2, or 3).

$\text{Ways to choose red} = 2^3 = 8$

2. Total Number of Groups

To find the total number of groups, we multiply the independent choices for each color:

\text{Total groups} = (\text{Ways for Green}) \times (\text{Ways for Blue}) \times (\text{Ways for Red})

Substitute the values:

\text{Total groups} = (2^5 - 1) \times (2^4 - 1) \times (2^3)

\text{Total groups} = 31 \times 15 \times 8

Calculating the final value:

31 \times 120 = 3720

Final Answer

The total number of groups that can be formed is 3720.