[Math] A box contains $5$ red balls, $3$ blue balls, and $2$ yellow balls such that all balls are different

combinatorics

Q) A box contains $5$ red balls, $3$ blue balls, and $2$ yellow balls. Assume that all balls are different. In how many ways can you select $4$ balls so that each selection contains:

  1. Exactly two red balls$\quad\dots\space {^5C_2} \times {^5C_2}\quad?$

  2. At least two red balls?

  3. No yellow balls

  4. At most two red balls

Best Answer

You are right with the first one

  1. C(5, 2) * C(5, 2)+C(5, 3)*C(5, 1)+C(5, 4)

You can select two red balls form the four red balls and two from the other remaining five balls. Similarly you can select three red balls and one front the remaining. Touch can also select four red balls.

  1. C(5, 4)

You can select any balls other than red.

4.C(5, 4)+ C(5, 1)*C(5, 3)+C(5, 2)*C(5, 2)

The selection can have no red balls , only one red ball or only two red balls

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