<s>[INST] A box has $4$ red and $20$ white balls. A person takes $10$ balls. What is the probability that all or none of the red balls were taken? A box has $24$ balls, $4$ red and $20$ white. One person takes $10$ balls and the second the remaining $14$. What is the probability that one of the two people picked up the $4$ red ones?
I don't understand why is this correct.
$$\frac{{20 \choose 6}+{20 \choose 10}}{24 \choose 10} $$
 [/INST] Because $\tfrac{\binom 44\binom{20}6\binom{14}{14}}{\binom{24}{10}\binom{14}{14}}$ is the probability that the first person select 4 red and 6 white and the second person receives the remaining 14 white, while $\tfrac{\binom{20}{10}\binom{4}{4}\binom{10}{10}}{\binom{24}{10}\binom{14}{14}}$ is the probability that the first person selects 10 white and second person select 4 red and 10 white.
$$~\\\dfrac{\binom 44\binom{20}6\binom{14}{14}+\binom{20}{10}\binom{4}{4}\binom{10}{10}}{\binom{24}{10}\binom{14}{14}}=\dfrac{\binom{20}6+\binom{20}{10}}{\binom{24}{10}}$$
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