The number of ways of choosing $3$ balls from $2$ yellow, $3$ green, and $5$ blue balls such that at least $2$ blue balls are always selected.

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Solution

There are $2$ Yellow balls $3$ Green balls and $5$ Blue balls
Possible cases
Case $1 :$
Choosing 2 blue balls and 1 yellow ball
Number of ways is $^{5} C_{2} \times^{2} C_{1} = 10 \times 2 = 20$

Case $2 :$
Choosing 2 blue balls and 1 green ball
Number of ways is $^{5} C_{2} \times^{3} C_{1} = 10 \times 3 = 30$

Case $3 :$
Choosing 3 blue balls
Number of ways is $^{5} C_{3} = 10$

Total number of ways are $20 + 30 + 10 = 60$

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The number of ways of choosing 3 balls from 2 yellow, 3 green, and 5 blue balls such that at least 2 blue balls are always selected.

The number of ways of choosing $3$ balls from $2$ yellow, $3$ green, and $5$ blue balls such that at least $2$ blue balls are always selected.