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Question Number 50258 by rahul 19 last updated on 15/Dec/18
$${A}\:{delegation}\:{of}\:\mathrm{4}\:{students}\:{is}\:{to}\:{be} \\ $$$${selected}\:{from}\:{a}\:{total}\:{of}\:\mathrm{12}\:{students}. \\ $$$${In}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{delegation} \\ $$$${be}\:{selected}\:{if}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{2}\:{particular}\:{students}\:{wish}\:{to}\:{be}\: \\ $$$${included}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{2}\:{particular}\:{students}\:{refuse}\:{to}\:{be}\: \\ $$$${together}\:{and}\:\mathrm{2}\:{other}\:{particular}\:{students} \\ $$$${wish}\:{to}\:{be}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$
Answered by mr W last updated on 15/Dec/18
$${A},{B}\:{are}\:{the}\:{two}\:{students}\:{who}\:{want} \\ $$$${to}\:{be}\:{together}\:{in}\:{the}\:{delegation}. \\ $$$${C},{D}\:{are}\:{the}\:{two}\:{students}\:{who}\:{refuse} \\ $$$${to}\:{be}\:{together}\:{in}\:{the}\:{delegation}. \\ $$$$\left.\mathrm{1}\right) \\ $$$${with}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{2}} ^{\mathrm{10}} \:{ways} \\ $$$${without}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{4}} ^{\mathrm{10}} \:{ways} \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{10}} +{C}_{\mathrm{4}} ^{\mathrm{10}} =\mathrm{255}\:{ways} \\ $$$$ \\ $$$$\left.\mathrm{2}\right) \\ $$$${with}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1} \\ $$$${to}\:{select}\:{the}\:{other}\:{two}\:{delegates}\:{from} \\ $$$${the}\:{other}\:\mathrm{10}\:{students},\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{10}} \\ $$$${ways},\:{but}\:{the}\:{selection}\:{with}\:{C}\:{and}\:{D} \\ $$$${together}\:{must}\:{be}\:{excluded},\:{therefore} \\ $$$${there}\:{are}\:{only}\:{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1}\:{ways}. \\ $$$$ \\ $$$${without}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} \\ $$$${all}\:\mathrm{4}\:{delegates}\:{will}\:{be}\:{selected}\:{from}\:{the} \\ $$$${other}\:\mathrm{10}\:{students},\:{there}\:{are}\:{C}_{\mathrm{4}} ^{\mathrm{10}} \:{ways}, \\ $$$${with}\:{C}\:{and}\:{D}\:{together}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{8}} \\ $$$${ways}\:{which}\:{must}\:{be}\:{excluded},\:{therefore} \\ $$$${there}\:{are}\:{only}\:{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} \:{ways}. \\ $$$$ \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1}+{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} =\mathrm{255}−\mathrm{29}=\mathrm{226}\:{ways} \\ $$
Commented by rahul 19 last updated on 15/Dec/18
thank you sir!��