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Permutation and CombinationQuestion and Answers: Page 3

Question Number 194638 Answers: 1 Comments: 1

Prove that ∀n∈IN^∗ Σ_(k=1) ^(2^n −1) (1/(sin^2 (((kπ)/2^(n+1) ))))= ((2^(2n+1) −2)/3) Give in terms of n Σ_(k=1) ^(2^n −1) (1/(sin^4 (((kπ)/2^(n+1) ))))

$$\mathrm{Prove}\:\mathrm{that}\:\forall{n}\in\mathrm{IN}^{\ast} \:\:\:\:\: \\ $$$$\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}} }\right)}=\:\frac{\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} −\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{Give}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}\:\:\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}^{{n}} −\mathrm{1}} {\sum}}\:\frac{\mathrm{1}}{{sin}^{\mathrm{4}} \left(\frac{{k}\pi}{\mathrm{2}^{{n}+\mathrm{1}} }\right)} \\ $$

Question Number 193864 Answers: 1 Comments: 0

Question Number 193368 Answers: 2 Comments: 0

If log_a y = (1/3) and log_8 a = x + 1 then show that y = 2^(x + 1)

$$\mathrm{If}\:\mathrm{log}_{{a}} {y}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{and}\:\mathrm{log}_{\mathrm{8}} {a}\:=\:{x}\:+\:\mathrm{1}\:\mathrm{then}\:\mathrm{show} \\ $$$$\mathrm{that}\:{y}\:=\:\mathrm{2}^{{x}\:+\:\mathrm{1}} \\ $$

Question Number 192960 Answers: 0 Comments: 0

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Question Number 192256 Answers: 0 Comments: 0

given f(x)=cx(x−20) and A=(2,5) find the nearst point to A on the graph

$$\mathrm{given}\:{f}\left({x}\right)={cx}\left({x}−\mathrm{20}\right)\:\mathrm{and}\:{A}=\left(\mathrm{2},\mathrm{5}\right) \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{nearst}\:\mathrm{point}\:\mathrm{to}\:{A}\:\mathrm{on}\:\mathrm{the}\:\mathrm{graph} \\ $$

Question Number 191934 Answers: 1 Comments: 0

Question Number 191277 Answers: 0 Comments: 0

Question Number 190738 Answers: 0 Comments: 0

∫_0 ^( (π/2)) (((sin((x/2^n )))/(sinx))) dx , n ∈ N

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\left(\frac{{sin}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)}{{sinx}}\right)\:{dx}\:,\:{n}\:\in\:\mathbb{N} \\ $$

Question Number 190602 Answers: 1 Comments: 0

let S={a,b,c,d,e,f} if we take any subset S (same subset is allowed), it also can be S, which will form S if we join them, order of operation does not matter ({a,b,c,d},{d,e,f}) is the same as ({d,e,f},{a,b,c,d}) how many ways can we choose?

$$ \\ $$$$\: \\ $$$$\:{let}\:{S}=\left\{{a},{b},{c},{d},{e},{f}\right\} \\ $$$$\:{if}\:{we}\:{take}\:{any}\:{subset}\:{S}\:\left({same}\:{subset}\:{is}\:{allowed}\right), \\ $$$$\:{it}\:{also}\:{can}\:{be}\:{S},\:{which}\:{will}\:{form}\:{S}\:{if}\:{we}\:{join}\:{them}, \\ $$$${order}\:{of}\:{operation}\:{does}\:{not}\:{matter} \\ $$$$\:\left(\left\{{a},{b},{c},{d}\right\},\left\{{d},{e},{f}\right\}\right)\:{is}\:{the}\:{same}\:{as} \\ $$$$\:\left(\left\{{d},{e},{f}\right\},\left\{{a},{b},{c},{d}\right\}\right) \\ $$$$\:{how}\:{many}\:{ways}\:{can}\:{we}\:{choose}? \\ $$$$\: \\ $$$$ \\ $$

Question Number 190347 Answers: 1 Comments: 0

how is solution lim_(x→sinπ ) ((sin(π/2))/(sinx))=?

$${how}\:{is}\:{solution} \\ $$$$\underset{{x}\rightarrow\mathrm{sin}\pi\:} {\mathrm{lim}}\frac{\mathrm{sin}\frac{\pi}{\mathrm{2}}}{\mathrm{sin}{x}}=? \\ $$

Question Number 190260 Answers: 1 Comments: 0

f : [1, 3] →R , f(x) = (1/x) A(1, 1) B(1, (1/3)) B′(b, (1/b)) , b ≥ 1 Find i. equation of line AB′ ii. equation of tangent T ′ to C_f at point with x = ((1 + b)/2) iii. Study relative positions of L_(AB ′) , T ′ to C_f

$${f}\::\:\left[\mathrm{1},\:\mathrm{3}\right]\:\rightarrow\mathbb{R}\:,\:{f}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}} \\ $$$${A}\left(\mathrm{1},\:\mathrm{1}\right) \\ $$$${B}\left(\mathrm{1},\:\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$${B}'\left({b},\:\frac{\mathrm{1}}{{b}}\right)\:,\:{b}\:\geqslant\:\mathrm{1} \\ $$$${Find} \\ $$$${i}.\:{equation}\:{of}\:{line}\:{AB}' \\ $$$${ii}.\:{equation}\:{of}\:{tangent}\:{T}\:'\:{to}\:{C}_{{f}} \:{at}\:{point} \\ $$$${with}\:{x}\:=\:\frac{\mathrm{1}\:+\:{b}}{\mathrm{2}} \\ $$$${iii}.\:{Study}\:{relative}\:{positions}\:{of}\:{L}_{{AB}\:'} \:,\:{T}\:'\:{to}\:{C}_{{f}} \\ $$

Question Number 189614 Answers: 1 Comments: 0

Question Number 189049 Answers: 1 Comments: 2

Question Number 189021 Answers: 2 Comments: 4

How many non−similar triangles have integer angles in °?

$${How}\:{many}\:{non}−{similar}\:{triangles} \\ $$$${have}\:{integer}\:{angles}\:{in}\:°? \\ $$

Question Number 188551 Answers: 1 Comments: 0

you randomly select a 5 digit number. what′s the probability that this number has exactly 3 different digits?

$${you}\:{randomly}\:{select}\:{a}\:\mathrm{5}\:{digit}\:{number}. \\ $$$${what}'{s}\:{the}\:{probability}\:{that}\:{this}\:{number} \\ $$$${has}\:{exactly}\:\mathrm{3}\:{different}\:{digits}? \\ $$

Question Number 188362 Answers: 2 Comments: 0

find the number of 5 digit natural numbers with strictly ascending digits whose sum is 20. example: 12458 is such a number

$${find}\:{the}\:{number}\:{of}\:\mathrm{5}\:{digit}\:{natural} \\ $$$${numbers}\:{with}\:{strictly}\:{ascending}\: \\ $$$${digits}\:{whose}\:{sum}\:{is}\:\mathrm{20}. \\ $$$${example}:\:\mathrm{12458}\:{is}\:{such}\:{a}\:{number} \\ $$

Question Number 188301 Answers: 1 Comments: 4

Find the number of triangles with integer side lengths and perimeter p.

$${Find}\:{the}\:{number}\:{of}\:{triangles}\:{with} \\ $$$${integer}\:{side}\:{lengths}\:{and}\:{perimeter}\:{p}. \\ $$

Question Number 187948 Answers: 1 Comments: 0

Given a set H={1,2,3,...,300. We will a create a subset of H consisting of three elements. If the sum of the three elements is divisible by 3 then the number of subsets that canbe made is x. Find the remainder if x is divided by 100000

$$ \\ $$$$\mathrm{Given}\:\mathrm{a}\:\mathrm{set}\:\mathrm{H}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{300}.\:\mathrm{We}\:\mathrm{will}\:\mathrm{a}\right. \\ $$$$\mathrm{cre}{a}\mathrm{te}\:\mathrm{a}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{H}\:\mathrm{consisting}\:\mathrm{of}\: \\ $$$$\mathrm{thre}{e}\:\mathrm{elements}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{thr}{e}\mathrm{e}\:\mathrm{elements}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}\:\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{subsets}\:\mathrm{that}\: \\ $$$$\mathrm{canbe}\:\mathrm{made}\:\mathrm{is}\:\mathrm{x}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{remaind}{e}\mathrm{r}\:\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{100000} \\ $$

Question Number 185659 Answers: 1 Comments: 1

prove for r, n ∈ N Σ_(k=r) ^n ((k),(r) ) = (((n+1)),((r+1)) ) (Hockey−stick identity)

$${prove}\:{for}\:{r},\:{n}\:\in\:\mathbb{N} \\ $$$$\underset{{k}={r}} {\overset{{n}} {\sum}}\begin{pmatrix}{{k}}\\{{r}}\end{pmatrix}\:=\begin{pmatrix}{{n}+\mathrm{1}}\\{{r}+\mathrm{1}}\end{pmatrix} \\ $$$$\left({Hockey}−{stick}\:{identity}\right) \\ $$

Question Number 185642 Answers: 1 Comments: 1

Question Number 185350 Answers: 0 Comments: 1

C_4 ^4 +C_4 ^5 +C_4 ^6 +...+C_4 ^(26) =?

$$\:\:\:{C}_{\mathrm{4}} ^{\mathrm{4}} +{C}_{\mathrm{4}} ^{\mathrm{5}} +{C}_{\mathrm{4}} ^{\mathrm{6}} +...+{C}_{\mathrm{4}} ^{\mathrm{26}} \:=? \\ $$

Question Number 182793 Answers: 0 Comments: 0

Suppose now that Allie is actually 70 years old, and that her life expectancy (if cured) is 12 years rather than 20 . Which procedure now has the greather expected value ? What if her life expectancy is 8 years?

$$\:{Suppose}\:{now}\:{that}\:{Allie}\:{is}\:{actually} \\ $$$$\mathrm{70}\:{years}\:{old},\:{and}\:{that}\:{her}\:{life}\:{expectancy} \\ $$$$\left({if}\:{cured}\right)\:{is}\:\mathrm{12}\:{years}\:{rather}\:{than} \\ $$$$\mathrm{20}\:.\:{Which}\:{procedure}\:{now}\:{has}\:{the} \\ $$$${greather}\:{expected}\:{value}\:?\:{What}\:{if}\:{her} \\ $$$${life}\:{expectancy}\:{is}\:\mathrm{8}\:{years}? \\ $$

Question Number 182695 Answers: 1 Comments: 2

If two events A and B are such that P(A^c )=0.3 ; P(B)=0.4 and P(A∩B^c )= 0.5 then P((B/(A∪B^c )))=? (A) 0.20 (B) 0.25 (C) 0.30 (D) 0.35

$$\:\:\mathrm{If}\:\mathrm{two}\:\mathrm{events}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{such}\: \\ $$$$\:\:\mathrm{that}\:\mathrm{P}\left(\mathrm{A}^{\mathrm{c}} \right)=\mathrm{0}.\mathrm{3}\:;\:\mathrm{P}\left(\mathrm{B}\right)=\mathrm{0}.\mathrm{4}\:\mathrm{and} \\ $$$$\:\:\mathrm{P}\left(\mathrm{A}\cap\mathrm{B}^{\mathrm{c}} \right)=\:\mathrm{0}.\mathrm{5}\:\mathrm{then}\:\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}\cup\mathrm{B}^{\mathrm{c}} }\right)=? \\ $$$$\:\left(\mathrm{A}\right)\:\mathrm{0}.\mathrm{20}\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{0}.\mathrm{25}\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{0}.\mathrm{30}\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{0}.\mathrm{35} \\ $$$$\:\: \\ $$

Question Number 181971 Answers: 1 Comments: 0

How many words can be formed from different letters of the “Your answer is wrong” with not repeating? So that they shouldn′t start with ′o′ neither ′w′ and shouldn′t end with ′w′, and should be′r′ & ′s′ adjacents. Oya...n, Wen...y, Iog...w , Yourws...e : are invalid Enrsow...g : is valid Q.180162

$${How}\:{many}\:{words}\:{can}\:{be}\:{formed}\:{from}\:{different} \\ $$$${letters}\:{of}\:{the}\:``{Your}\:{answer}\:{is}\:{wrong}''\:{with}\:{not} \\ $$$${repeating}?\:{So}\:{that}\:{they}\:{shouldn}'{t}\:{start}\:{with}\:'{o}'\: \\ $$$$\:{neither}\:'{w}'\:{and}\:{shouldn}'{t}\:{end}\:{with}\:'{w}',\:{and} \\ $$$$\:{should}\:{be}'{r}'\:\&\:'{s}'\:{adjacents}. \\ $$$$ \\ $$$$\:\cancel{{O}ya}...{n},\:\cancel{{W}en}...{y},\:{Iog}...\cancel{{w}}\:,\:{You}\cancel{{r}w}\cancel{{s}}...{e}\:\::\:{are}\:{invalid} \\ $$$$\:{Enrsow}...{g}\::\:{is}\:{valid} \\ $$$${Q}.\mathrm{180162} \\ $$

Question Number 181939 Answers: 2 Comments: 0

If 10 different balls are to be placed in 4 boxes at random , then the probability that two of these boxes contain exactly 2 and 3 balls

$${If}\:\mathrm{10}\:{different}\:{balls}\:{are}\:{to}\:{be}\:{placed} \\ $$$${in}\:\mathrm{4}\:{boxes}\:{at}\:{random}\:,\:{then}\:{the}\:{probability} \\ $$$${that}\:{two}\:{of}\:{these}\:{boxes}\:{contain} \\ $$$${exactly}\:\mathrm{2}\:{and}\:\mathrm{3}\:{balls}\: \\ $$

Question Number 181251 Answers: 1 Comments: 1

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