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Question Number 102800 by bramlex last updated on 11/Jul/20
There are 14 boys and 10 girls in  a classroom. The teacher wants  to form a team of 5 students .  The team must have a least two  boys and two girls . Find the  number of ways the team can be  chosen.
$$\mathrm{There}\:\mathrm{are}\:\mathrm{14}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{10}\:\mathrm{girls}\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{classroom}.\:\mathrm{The}\:\mathrm{teacher}\:\mathrm{wants} \\ $$$$\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{team}\:\mathrm{of}\:\mathrm{5}\:\mathrm{students}\:. \\ $$$$\mathrm{The}\:\mathrm{team}\:\mathrm{must}\:\mathrm{have}\:\mathrm{a}\:\mathrm{least}\:\mathrm{two} \\ $$$$\mathrm{boys}\:\mathrm{and}\:\mathrm{two}\:\mathrm{girls}\:.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{the}\:\mathrm{team}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{chosen}.\: \\ $$
Answered by bemath last updated on 11/Jul/20
case(1) 3boys &2 girls⇒C_3 ^(14)  ×C_2 ^(10)   =((14.13.12)/(3.2.1)) × ((10.9)/(2.1)) = 13680  (2)2boys &3 girls⇒C_2 ^(14) ×C_3 ^(10)  = ((14.13)/(2.1))×((10.9.8)/(3.2.1))  = 10920  totally = 13680+10920 =24600
$${case}\left(\mathrm{1}\right)\:\mathrm{3}{boys}\:\&\mathrm{2}\:{girls}\Rightarrow{C}_{\mathrm{3}} ^{\mathrm{14}} \:×{C}_{\mathrm{2}} ^{\mathrm{10}} \\ $$$$=\frac{\mathrm{14}.\mathrm{13}.\mathrm{12}}{\mathrm{3}.\mathrm{2}.\mathrm{1}}\:×\:\frac{\mathrm{10}.\mathrm{9}}{\mathrm{2}.\mathrm{1}}\:=\:\mathrm{13680} \\ $$$$\left(\mathrm{2}\right)\mathrm{2}{boys}\:\&\mathrm{3}\:{girls}\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{14}} ×{C}_{\mathrm{3}} ^{\mathrm{10}} \:=\:\frac{\mathrm{14}.\mathrm{13}}{\mathrm{2}.\mathrm{1}}×\frac{\mathrm{10}.\mathrm{9}.\mathrm{8}}{\mathrm{3}.\mathrm{2}.\mathrm{1}} \\ $$$$=\:\mathrm{10920} \\ $$$${totally}\:=\:\mathrm{13680}+\mathrm{10920}\:=\mathrm{24600} \\ $$$$ \\ $$

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