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Question: A school consists of 7 teachers of whom 4 are males and 3 are females .A committee of 2 teachers is to be formed . What is the probability that no male teachers will be in the committee ? Why the answer is $\frac{3C2}{7C2}$ (All committee members are female) not $\frac{4C0}{7C2}$ (No committee member is male) ?

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  • $\begingroup$ How did you get the second answer? Are you thinking that since no males are chosen, the answer is four choose zero? $\endgroup$ Aug 3, 2017 at 0:22
  • $\begingroup$ Yes I was thinking that. $\endgroup$
    – Utshaw
    Aug 3, 2017 at 0:26
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    $\begingroup$ Well, it's very good that no males are chosen. But then, you still have to make a choice among the females, right? In truth, in the first formula, the fact that no males are chosen is hidden, indeed it should also be included there, as the answers below point out. Because no males are chosen, there's no need to make a choice amongst males as well. $\endgroup$ Aug 3, 2017 at 0:34

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The answer is properly ${{^4C_0}\,{^3C_2}}/{^7C_2}$, the probability for selecting none from four and two from three when selecting two from seven.   However, since $^4C_0$ equals $1$ that simplifies to just: $$\dfrac{^3C_2}{^7C_2}$$

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If you have to choose $2$ our of $4$ female teachers, then you have to choose $0$ out of $4$ males, namely, $\binom{4}{0}$ males. And then multiply it by the total number of possible $2$ out of $3$ females, namely $\binom{3}{2}$, i.e., $$ \frac{\binom{4}{0} \binom{3}{2}}{\binom{7}{2}} $$

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