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Question Number 135490 by benjo_mathlover last updated on 13/Mar/21

$$Combination$$ A committee of 8 people is to be formed from 7 women and 5 men. In how many ways can the members be chosen so as to include at least 3 men?\n

Answered by EDWIN88 last updated on 13/Mar/21

$$=\left(\begin{array}{c}5\\ 3\end{array}\right)\left(\begin{array}{c}7\\ 5\end{array}\right)+\left(\begin{array}{c}5\\ 4\end{array}\right)\left(\begin{array}{c}7\\ 4\end{array}\right)+\left(\begin{array}{c}5\\ 3\end{array}\right)\left(\begin{array}{c}7\\ 3\end{array}\right)=420$$

Answered by mr W last updated on 13/Mar/21

$$thecoef.ofterm{x}^{8}in$$ $$(C_3^5{x}^3+C_4^5{x}^4+C_5^5{x}^5)(1+C_1^{7}x+C_2^7{x}^2+...+C_7^7{x}^7)$$ $$is420.thatmeansthereare420ways.$$

Commented bymr W last updated on 13/Mar/21

Answered by physicstutes last updated on 13/Mar/21

$$\mathrm{i}\mathrm{like}\mathrm{looking}\mathrm{at}\mathrm{it}\mathrm{from}\mathrm{this}\mathrm{pespective}.\mathrm{see}\mathrm{the}\mathrm{table}\mathrm{below}$$ $$\mathrm{which}\mathrm{shows}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{men}\mathrm{and}\mathrm{women}\mathrm{selected}\mathrm{based}\mathrm{on}$$ $$\mathrm{the}\mathrm{above}\mathrm{conditions}$$ $$\begin{array}{|cc|}\hline \mathrm{number}\mathrm{of}\mathrm{men}& \mathrm{number}\mathrm{of}\mathrm{women}\\ 3& 5\\ 4& 4\\ 5& 3\\ \hline\end{array}$$ $$\mathrm{therefore}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{ways}={}^5{C}_3{\times }^8{C}_5+{}^5{C}_4{\times }^8{C}_4+{}^5{C}_5{\times }^5{C}_3=420\mathrm{ways}$$

Commented bymr W last updated on 13/Mar/21

$$inthiscaseyoucanusethismethod.$$ $$butforcomplexcasesforexample$$ $$thereare10men,12women,8boys,$$ $$7girls.acommitteewith10persons$$ $$shouldbeformed,weneedother$$ $$methods.$$

Commented byphysicstutes last updated on 13/Mar/21

$$\mathrm{sure}\mathrm{sir}\mathrm{i}\mathrm{totally}\mathrm{know}\mathrm{that}.$$

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