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A-delegation-of-4-students-is-to-be-selected-from-a-total-of-12-students-In-how-many-ways-can-the-delegation-be-selected-if-1-2-particular-students-wish-to-be-included-together-only-in-the-delegat




Question Number 50258 by rahul 19 last updated on 15/Dec/18
A delegation of 4 students is to be  selected from a total of 12 students.  In how many ways can the delegation  be selected if:  1) 2 particular students wish to be   included together only in the delegation?  2) 2 particular students refuse to be   together and 2 other particular students  wish to be together only in the delegation?
$${A}\:{delegation}\:{of}\:\mathrm{4}\:{students}\:{is}\:{to}\:{be} \\ $$$${selected}\:{from}\:{a}\:{total}\:{of}\:\mathrm{12}\:{students}. \\ $$$${In}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{delegation} \\ $$$${be}\:{selected}\:{if}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{2}\:{particular}\:{students}\:{wish}\:{to}\:{be}\: \\ $$$${included}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{2}\:{particular}\:{students}\:{refuse}\:{to}\:{be}\: \\ $$$${together}\:{and}\:\mathrm{2}\:{other}\:{particular}\:{students} \\ $$$${wish}\:{to}\:{be}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$
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.  1)  with A and B: C_2 ^(10)  ways  without A and B: C_4 ^(10)  ways  ⇒C_2 ^(10) +C_4 ^(10) =255 ways    2)  with A and B: C_2 ^(10) −1  to select the other two delegates from  the other 10 students, there are C_2 ^(10)   ways, but the selection with C and D  together must be excluded, therefore  there are only C_2 ^(10) −1 ways.    without A and B: C_4 ^(10) −C_2 ^8   all 4 delegates will be selected from the  other 10 students, there are C_4 ^(10)  ways,  with C and D together there are C_2 ^8   ways which must be excluded, therefore  there are only C_4 ^(10) −C_2 ^8  ways.    ⇒C_2 ^(10) −1+C_4 ^(10) −C_2 ^8 =255−29=226 ways
$${A},{B}\:{are}\:{the}\:{two}\:{students}\:{who}\:{want} \\ $$$${to}\:{be}\:{together}\:{in}\:{the}\:{delegation}. \\ $$$${C},{D}\:{are}\:{the}\:{two}\:{students}\:{who}\:{refuse} \\ $$$${to}\:{be}\:{together}\:{in}\:{the}\:{delegation}. \\ $$$$\left.\mathrm{1}\right) \\ $$$${with}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{2}} ^{\mathrm{10}} \:{ways} \\ $$$${without}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{4}} ^{\mathrm{10}} \:{ways} \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{10}} +{C}_{\mathrm{4}} ^{\mathrm{10}} =\mathrm{255}\:{ways} \\ $$$$ \\ $$$$\left.\mathrm{2}\right) \\ $$$${with}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1} \\ $$$${to}\:{select}\:{the}\:{other}\:{two}\:{delegates}\:{from} \\ $$$${the}\:{other}\:\mathrm{10}\:{students},\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{10}} \\ $$$${ways},\:{but}\:{the}\:{selection}\:{with}\:{C}\:{and}\:{D} \\ $$$${together}\:{must}\:{be}\:{excluded},\:{therefore} \\ $$$${there}\:{are}\:{only}\:{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1}\:{ways}. \\ $$$$ \\ $$$${without}\:{A}\:{and}\:{B}:\:{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} \\ $$$${all}\:\mathrm{4}\:{delegates}\:{will}\:{be}\:{selected}\:{from}\:{the} \\ $$$${other}\:\mathrm{10}\:{students},\:{there}\:{are}\:{C}_{\mathrm{4}} ^{\mathrm{10}} \:{ways}, \\ $$$${with}\:{C}\:{and}\:{D}\:{together}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{8}} \\ $$$${ways}\:{which}\:{must}\:{be}\:{excluded},\:{therefore} \\ $$$${there}\:{are}\:{only}\:{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} \:{ways}. \\ $$$$ \\ $$$$\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{10}} −\mathrm{1}+{C}_{\mathrm{4}} ^{\mathrm{10}} −{C}_{\mathrm{2}} ^{\mathrm{8}} =\mathrm{255}−\mathrm{29}=\mathrm{226}\:{ways} \\ $$
Commented by rahul 19 last updated on 15/Dec/18
thank you sir!��

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