The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 prof…

Question

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

Accepted Answer

$\begin{gathered} 3E, \quad  1D, \quad  8P \ 3E, \quad  2D, \quad  7P \ 4E, \quad  1D, \quad  7P \ 4E, \quad  2D, \quad  6P \end{gathered}$

$\begin{gathered} P=\frac{{}^{4}{C}_{3}\cdot {}^{2}{C}_{1}\cdot {}^{10}{C}_{8}+{}^{4}{C}_{3}\cdot {}^{2}{C}_{2}\cdot {}^{10}{C}_{7}+{}^{4}{C}_{4}\cdot {}^{2}{C}_{1}\cdot {}^{10}{C}_{7}+{}^{4}{C}_{4}\cdot {}^{2}{C}_{2}\cdot {}^{10}{C}_{6}}{{}^{10}{C}_{12}} \ =\frac{129}{182} \end{gathered}$

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