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Permutation and CombinationQuestion and Answers: Page 1

Question Number 222192 Answers: 1 Comments: 0

there are 32 students in a class. for each competition in a sport event in the school each class can send a team with three students. if no two students may be in the same team for more than one time, in how many different competitions can this class participate?

$${there}\:{are}\:\mathrm{32}\:{students}\:{in}\:{a}\:{class}.\:{for} \\ $$$${each}\:{competition}\:{in}\:{a}\:{sport}\:{event}\: \\ $$$${in}\:{the}\:{school}\:{each}\:{class}\:{can}\:{send} \\ $$$${a}\:{team}\:{with}\:{three}\:{students}.\:{if}\:{no} \\ $$$${two}\:{students}\:{may}\:{be}\:{in}\:{the}\:{same} \\ $$$${team}\:{for}\:{more}\:{than}\:{one}\:{time},\:{in} \\ $$$${how}\:{many}\:{different}\:{competitions}\: \\ $$$${can}\:{this}\:{class}\:{participate}? \\ $$

Question Number 222105 Answers: 1 Comments: 1

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Question Number 219936 Answers: 1 Comments: 0

Prove that:∀n∈IN ∫^( n+1) _( n) ln(t)dt≤ln(n+(1/2))

$$\mathrm{Prove}\:\mathrm{that}:\forall\mathrm{n}\in\mathrm{IN} \\ $$$$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} \mathrm{ln}\left(\mathrm{t}\right)\mathrm{dt}\leqslant\mathrm{ln}\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 219527 Answers: 2 Comments: 2

Question Number 218578 Answers: 2 Comments: 0

10 couples are invited to a dinner. after the dinner they form pairs to dance. in how many ways can they do this, 1) generally 2) a man should dance with a woman 3) as 2), but two special couples should not dance with each other 4) a man should dance with a woman, but not with his own wife 5) as 4), but two special couples should not dance with each other

$$\mathrm{10}\:{couples}\:{are}\:{invited}\:{to}\:{a}\:{dinner}. \\ $$$${after}\:{the}\:{dinner}\:{they}\:{form}\:{pairs}\: \\ $$$${to}\:{dance}.\:{in}\:{how}\:{many}\:{ways}\:{can}\:{they} \\ $$$${do}\:{this}, \\ $$$$\left.\mathrm{1}\right)\:{generally} \\ $$$$\left.\mathrm{2}\right)\:{a}\:{man}\:{should}\:{dance}\:{with}\:{a}\:{woman} \\ $$$$\left.\mathrm{3}\left.\right)\:{as}\:\mathrm{2}\right),\:{but}\:{two}\:{special}\:{couples} \\ $$$$\:\:\:\:\:{should}\:{not}\:{dance}\:{with}\:{each}\:{other} \\ $$$$\left.\mathrm{4}\right)\:{a}\:{man}\:{should}\:{dance}\:{with}\:{a}\:{woman}, \\ $$$$\:\:\:\:\:{but}\:{not}\:{with}\:{his}\:{own}\:{wife} \\ $$$$\left.\mathrm{5}\left.\right)\:{as}\:\mathrm{4}\right),\:{but}\:{two}\:{special}\:{couples} \\ $$$$\:\:\:\:\:{should}\:{not}\:{dance}\:{with}\:{each}\:{other} \\ $$

Question Number 218445 Answers: 3 Comments: 0

there are 100 students in a school. it is found out that each student should select at least 4 courses, so that no two students have the same selection. how many different courses does the school offer?

$${there}\:{are}\:\mathrm{100}\:{students}\:{in}\:{a}\:{school}. \\ $$$${it}\:{is}\:{found}\:{out}\:{that}\:{each}\:{student}\: \\ $$$${should}\:{select}\:{at}\:{least}\:\mathrm{4}\:{courses},\:{so}\: \\ $$$${that}\:{no}\:{two}\:{students}\:{have}\:{the}\:{same}\: \\ $$$${selection}.\: \\ $$$${how}\:{many}\:{different}\:{courses}\:{does}\: \\ $$$${the}\:{school}\:{offer}? \\ $$

Question Number 218310 Answers: 2 Comments: 1

10 couples are invited to a dinner and should be seated at a round table. in how many ways can the host do this, 1) generally. 2) if the husband and the wife of each couple should sit together. 3) if no two men should sit next to each other. 4) as 3), but two speicial couples should not sit next to each other. it means that the husband from couple 1 may not sit next to the wife from couple 2 and the husband from couple 2 may not sit next to the wife from couple 1. but certainly the husbands of both couples may sit next to their own wifes.

$$\mathrm{10}\:{couples}\:{are}\:{invited}\:{to}\:{a}\:{dinner} \\ $$$${and}\:{should}\:{be}\:{seated}\:{at}\:{a}\:{round}\:{table}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{host}\:{do} \\ $$$${this}, \\ $$$$\left.\mathrm{1}\right)\:{generally}. \\ $$$$\left.\mathrm{2}\right)\:{if}\:{the}\:{husband}\:{and}\:{the}\:{wife}\:{of} \\ $$$$\:\:\:\:\:{each}\:{couple}\:{should}\:{sit}\:{together}. \\ $$$$\left.\mathrm{3}\right)\:{if}\:{no}\:{two}\:{men}\:{should}\:{sit}\:{next} \\ $$$$\:\:\:\:\:{to}\:{each}\:{other}. \\ $$$$\left.\mathrm{4}\left.\right)\:{as}\:\mathrm{3}\right),\:{but}\:{two}\:{speicial}\:{couples}\: \\ $$$$\:\:\:\:\:{should}\:{not}\:{sit}\:{next}\:{to}\:{each}\:{other}. \\ $$$$\:\:\:\:\underline{\:{it}\:{means}\:}{that}\:{the}\:{husband}\:{from}\: \\ $$$$\:\:\:\:\:{couple}\:\mathrm{1}\:{may}\:{not}\:{sit}\:{next}\:{to}\:{the} \\ $$$$\:\:\:\:\:{wife}\:{from}\:{couple}\:\mathrm{2}\:{and}\:{the} \\ $$$$\:\:\:\:\:\:{husband}\:{from}\:{couple}\:\mathrm{2}\:{may}\:{not} \\ $$$$\:\:\:\:\:\:{sit}\:{next}\:{to}\:{the}\:{wife}\:{from}\:{couple}\:\mathrm{1}. \\ $$$$\:\:\:\:\:\:{but}\:{certainly}\:{the}\:{husbands}\:{of} \\ $$$$\:\:\:\:\:\:{both}\:{couples}\:{may}\:{sit}\:{next}\:{to}\:{their} \\ $$$$\:\:\:\:\:\:{own}\:{wifes}. \\ $$

Question Number 218169 Answers: 0 Comments: 0

P(5,6)=((15!)/((15−6))) = ((15!)/(9!)) =((15×14×131×2×11×10×9×8×7×6×5×4×3×2×1)/(9×8×7×6×5×4×3×2×)) =15×14×13×12×11×10 =3,603,600

$${P}\left(\mathrm{5},\mathrm{6}\right)=\frac{\mathrm{15}!}{\left(\mathrm{15}−\mathrm{6}\right)}\:=\:\frac{\mathrm{15}!}{\mathrm{9}!}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{15}×\mathrm{14}×\mathrm{131}×\mathrm{2}×\mathrm{11}×\mathrm{10}×\mathrm{9}×\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}}{\mathrm{9}×\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{15}×\mathrm{14}×\mathrm{13}×\mathrm{12}×\mathrm{11}×\mathrm{10} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3},\mathrm{603},\mathrm{600} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 218129 Answers: 2 Comments: 0

how many different words can be formed from the word MATHEMATICS? note: here a word should have at least two letters, but mustn′t have a meaning.

$${how}\:{many}\:{different}\:{words}\:{can}\:{be} \\ $$$${formed}\:{from}\:{the}\:{word}\: \\ $$$$\boldsymbol{\mathrm{MATHEMATICS}}? \\ $$$${note}:\:\:{here}\:{a}\:{word}\:{should}\:{have}\:{at}\: \\ $$$${least}\:{two}\:{letters},\:{but}\:{mustn}'{t}\:{have}\:{a} \\ $$$${meaning}. \\ $$

Question Number 217402 Answers: 3 Comments: 0