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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 218438 Answers: 1 Comments: 0

An amazing thing i saw S = 1 + 2 + 3 + 4 + 5 + 6... = 1 + 2(2/2 + 3/2 + 4/2 + 5/2 +6/2....) = 1 + 2(1 + 3/2 + 2 + 5/2 + 3...) = 1 + 2(1+ 2 + 3 ... + 3/2 + 5/2...) = 1 + 2S + 2Σ_(n= 1) ^∞ ((2n + 1)/2) or,S − 2S = 1 + Σ_(n=1) ^∞ 2n + 1 ∴ −S = Σ_(n=0) ^∞ 2n + 1 Sum of all odd numbers! I know the step S−2S = −S is not allowed

$${An}\:{amazing}\:{thing}\:{i}\:{saw} \\ $$$${S}\:=\:\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:\mathrm{5}\:+\:\mathrm{6}... \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{2}/\mathrm{2}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{4}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\mathrm{6}/\mathrm{2}....\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}\:+\:\mathrm{3}...\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}\left(\mathrm{1}+\:\mathrm{2}\:+\:\mathrm{3}\:...\:+\:\mathrm{3}/\mathrm{2}\:+\:\mathrm{5}/\mathrm{2}...\right) \\ $$$$\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{2}{S}\:+\:\mathrm{2}\underset{{n}=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{2}{n}\:+\:\mathrm{1}}{\mathrm{2}} \\ $$$${or},{S}\:−\:\mathrm{2}{S}\:=\:\mathrm{1}\:+\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{2}{n}\:+\:\mathrm{1} \\ $$$$ \\ $$$$\therefore\:−{S}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{2}{n}\:+\:\mathrm{1} \\ $$$${Sum}\:{of}\:{all}\:{odd}\:{numbers}! \\ $$$${I}\:{know}\:{the}\:{step}\:{S}−\mathrm{2}{S}\:=\:−{S}\:{is}\:{not}\:{allowed} \\ $$

Question Number 218437 Answers: 0 Comments: 0

Question Number 218428 Answers: 3 Comments: 0

Question Number 218427 Answers: 1 Comments: 0

give a recurrence relation for I_n . I_n =∫_0 ^1 (1/((1+x^2 )^n ))dx, ∀n ∈ N.

$$ \\ $$$${give}\:{a}\:{recurrence}\:{relation}\:{for}\:{I}_{{n}} . \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx},\:\forall{n}\:\in\:\mathbb{N}. \\ $$

Question Number 218421 Answers: 1 Comments: 3

prove ∫_( ∂𝚺) 𝛚=∫_( 𝚺 ) d𝛚

$$\mathrm{prove}\: \\ $$$$\int_{\:\partial\boldsymbol{\Sigma}} \:\boldsymbol{\omega}=\int_{\:\boldsymbol{\Sigma}\:} \mathrm{d}\boldsymbol{\omega} \\ $$

Question Number 218420 Answers: 0 Comments: 0

Question Number 218418 Answers: 1 Comments: 0

Question Number 218411 Answers: 2 Comments: 0

Question Number 218410 Answers: 3 Comments: 0

Question Number 218400 Answers: 4 Comments: 0

Hard problem..... prove. for all α∈Z α^(37) ≡α Mod(1729) pls help :(

$$\mathrm{Hard}\:\mathrm{problem}..... \\ $$$$\:\mathrm{prove}. \\ $$$$\:\mathrm{for}\:\mathrm{all}\:\alpha\in\mathbb{Z} \\ $$$$\alpha^{\mathrm{37}} \equiv\alpha\:\mathrm{Mod}\left(\mathrm{1729}\right) \\ $$$$\mathrm{pls}\:\mathrm{help}\::\left(\right. \\ $$

Question Number 218399 Answers: 1 Comments: 0

Question Number 218398 Answers: 1 Comments: 0

Question Number 218397 Answers: 0 Comments: 0

Question Number 218396 Answers: 0 Comments: 0

Question Number 218395 Answers: 0 Comments: 0

Question Number 218393 Answers: 1 Comments: 0

let ABC be a triangle with incenter I. prove that Ia . Ib . Ic ≤ ((abc)/8)

$$ \\ $$$$\:{let}\:{ABC}\:{be}\:{a}\:{triangle}\:{with}\:{incenter}\:{I}. \\ $$$$\:{prove}\:{that}\:{Ia}\:.\:{Ib}\:.\:{Ic}\:\:\leqslant\:\frac{{abc}}{\mathrm{8}}\:\: \\ $$$$ \\ $$

Question Number 218383 Answers: 0 Comments: 2

Question Number 218385 Answers: 3 Comments: 0

Question Number 218384 Answers: 3 Comments: 0

Question Number 218388 Answers: 0 Comments: 0

Question Number 218376 Answers: 0 Comments: 0

∫_0 ^∞ ((J_𝛂 (ar))/((r^2 +k^2 )𝛍))dr

$$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{J}}_{\boldsymbol{\alpha}} \left(\boldsymbol{{ar}}\right)}{\left(\boldsymbol{{r}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} \right)\boldsymbol{\mu}}\boldsymbol{{dr}}\: \\ $$$$ \\ $$

Question Number 218375 Answers: 1 Comments: 0

∫((√(tan x))/(sin^3 x cos x))dx

$$ \\ $$$$\:\:\int\frac{\sqrt{\boldsymbol{{tan}}\:\boldsymbol{{x}}}}{\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 218374 Answers: 0 Comments: 0

∫_0 ^∞ J_α ((√(ar)))e^(−r) dr

$$\: \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \boldsymbol{{J}}_{\alpha} \left(\sqrt{\boldsymbol{{ar}}}\right)\boldsymbol{{e}}^{−\boldsymbol{{r}}} \boldsymbol{{dr}}\: \\ $$$$\: \\ $$

Question Number 218366 Answers: 0 Comments: 2

Question Number 218365 Answers: 1 Comments: 0

Find: 𝛀 =lim_(n→∞) Σ_(k=1) ^n [ Σ_(i=1) ^k i (k − i + (1/3))]^(−1) = ?

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left[\:\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\:\mathrm{i}\:\left(\mathrm{k}\:−\:\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\right)\right]^{−\mathrm{1}} =\:? \\ $$

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