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Question Number 21874 Answers: 2 Comments: 0

Find the remainder if 2^(2006) is divided by 17

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:\:\:\mathrm{2}^{\mathrm{2006}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{17} \\ $$

Question Number 21870 Answers: 1 Comments: 0

A wire of mass 9.8 × 10^(−3) kg per meter passes over a frictionless pulley fixed on the top of an inclined frictionless plane which makes an angle of 30° with the horizontal. Masses M_1 and M_2 are tied at the two ends of the wire. The mass M_1 rests on the plane and the mass M_2 hangs freely vertically downward. The whole system is in equilibrium. Now a transverse wave propagates along the wire with a velocity of 100 m/s. Find M_1 and M_2 (g = 9.8 m/s^2 ).

$$\mathrm{A}\:\mathrm{wire}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{9}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{3}} \:\mathrm{kg}\:\mathrm{per}\:\mathrm{meter} \\ $$$$\mathrm{passes}\:\mathrm{over}\:\mathrm{a}\:\mathrm{frictionless}\:\mathrm{pulley}\:\mathrm{fixed} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{an}\:\mathrm{inclined}\:\mathrm{frictionless} \\ $$$$\mathrm{plane}\:\mathrm{which}\:\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{30}°\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{Masses}\:{M}_{\mathrm{1}} \:\mathrm{and}\:{M}_{\mathrm{2}} \:\mathrm{are} \\ $$$$\mathrm{tied}\:\mathrm{at}\:\mathrm{the}\:\mathrm{two}\:\mathrm{ends}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}.\:\mathrm{The} \\ $$$$\mathrm{mass}\:{M}_{\mathrm{1}} \:\mathrm{rests}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{mass}\:{M}_{\mathrm{2}} \:\mathrm{hangs}\:\mathrm{freely}\:\mathrm{vertically} \\ $$$$\mathrm{downward}.\:\mathrm{The}\:\mathrm{whole}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{transverse}\:\mathrm{wave} \\ $$$$\mathrm{propagates}\:\mathrm{along}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{100}\:\mathrm{m}/\mathrm{s}.\:\mathrm{Find}\:{M}_{\mathrm{1}} \:\mathrm{and}\:{M}_{\mathrm{2}} \\ $$$$\left({g}\:=\:\mathrm{9}.\mathrm{8}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right). \\ $$

Question Number 21840 Answers: 1 Comments: 0

A cone is placed inside a sphere. If volume of the cone is maximum, find the ratio of radius from the cone and sphere

$$\mathrm{A}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{inside}\:\mathrm{a}\:\mathrm{sphere}. \\ $$$$\mathrm{If}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{maximum}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{from}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{and}\:\mathrm{sphere} \\ $$

Question Number 21837 Answers: 0 Comments: 4

Question Number 21825 Answers: 0 Comments: 2

Find the simplest form of Σ_(k = 1) ^n 2^k [sin^2 (((2kπ)/3)) + (1/4)]

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{simplest}\:\mathrm{form}\:\mathrm{of} \\ $$$$\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{2}^{{k}} \left[\mathrm{sin}^{\mathrm{2}} \:\left(\frac{\mathrm{2}{k}\pi}{\mathrm{3}}\right)\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right] \\ $$

Question Number 21964 Answers: 1 Comments: 0

Question Number 21965 Answers: 1 Comments: 0

integrate ∫sin^3 xdx

$${integrate} \\ $$$$\int{sin}^{\mathrm{3}} {xdx} \\ $$

Question Number 22051 Answers: 0 Comments: 3

I have recently seen a different notation for integration, written as: ∫dxf(x) e.g. ∫dx(x+1)^2 Is this the same as: ∫f(x)dx ⇒ =∫(x+1)^2 dx ???

$$\mathrm{I}\:\mathrm{have}\:\mathrm{recently}\:\mathrm{seen}\:\mathrm{a}\:\mathrm{different}\:\mathrm{notation} \\ $$$$\mathrm{for}\:\mathrm{integration},\:\mathrm{written}\:\mathrm{as}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int{dxf}\left({x}\right) \\ $$$${e}.{g}. \\ $$$$\:\:\:\:\:\:\int{dx}\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{this}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int{f}\left({x}\right){dx} \\ $$$$\Rightarrow\:\:\:\:=\int\left({x}+\mathrm{1}\right)^{\mathrm{2}} {dx} \\ $$$$??? \\ $$

Question Number 21819 Answers: 0 Comments: 3

Consider the situation shown in figure in which a block P of mass 2 kg is placed over a block Q of mass 4 kg. The combination of the blocks are placed on inclined plane of inclination 37° with horizontal. The coefficient of friction between Block Q and inclined plane is μ_2 and in between the two blocks is μ_1 . The system is released from rest, then when will be the frictional force acting between the block is zero? (p) μ_1 = 0.4; μ_2 = 0 (q) μ_1 = 0.8; μ_2 = 0.8 (r) μ_1 = 0.4; μ_2 = 0.5 (s) μ_1 = 0.5; μ_2 = 0.4 (t) μ_1 = 0; μ_2 = 0.4

Question Number 21867 Answers: 0 Comments: 0

The number of points in the cartesian plane with integral coordinates satisfying the inequalities ∣x∣ ≤ 4, ∣y∣ ≤ 4 and ∣x − y∣ ≤ 4 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{points}\:\mathrm{in}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coordinates} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{inequalities}\:\mid{x}\mid\:\leqslant\:\mathrm{4},\:\mid{y}\mid\:\leqslant \\ $$$$\mathrm{4}\:\mathrm{and}\:\mid{x}\:−\:{y}\mid\:\leqslant\:\mathrm{4}\:\mathrm{is} \\ $$

Question Number 21815 Answers: 0 Comments: 2

sin18^0 =

$${sin}\mathrm{18}^{\mathrm{0}} = \\ $$

Question Number 21813 Answers: 0 Comments: 0

x_1 +x_2 +x_3 =5 ways=5−1C_(3−1) =4C_2 =6

$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} =\mathrm{5} \\ $$$${ways}=\mathrm{5}−\mathrm{1}{C}_{\mathrm{3}−\mathrm{1}} =\mathrm{4}{C}_{\mathrm{2}} =\mathrm{6} \\ $$

Question Number 21811 Answers: 0 Comments: 1

a_1 =1, a_(n+1) =(a_n /(√(a_n +n+1))) Σ_(n=1) ^∞ a_n =?

$${a}_{\mathrm{1}} =\mathrm{1},\:{a}_{{n}+\mathrm{1}} =\frac{{a}_{{n}} }{\sqrt{{a}_{{n}} +{n}+\mathrm{1}}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} =? \\ $$

Question Number 21810 Answers: 1 Comments: 0

integrate ∫(x^2 /(√(1−x^2 )))dx

$${integrate} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 21802 Answers: 1 Comments: 1

Five balls are to be placed in three boxes. Each can hold all the five balls. In how many different ways can we place the balls so that no box remains empty, if balls are different but boxes are identical?

$$\mathrm{Five}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{three} \\ $$$$\mathrm{boxes}.\:\mathrm{Each}\:\mathrm{can}\:\mathrm{hold}\:\mathrm{all}\:\mathrm{the}\:\mathrm{five}\:\mathrm{balls}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{we} \\ $$$$\mathrm{place}\:\mathrm{the}\:\mathrm{balls}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{box}\:\mathrm{remains} \\ $$$$\mathrm{empty},\:\mathrm{if}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{boxes} \\ $$$$\mathrm{are}\:\mathrm{identical}? \\ $$

Question Number 21801 Answers: 0 Comments: 0

There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their point of intersection are joined. Then the number of fresh lines thus obtained is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane},\:\mathrm{no} \\ $$$$\mathrm{two}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{and}\:\mathrm{no}\:\mathrm{three} \\ $$$$\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{same}\:\mathrm{point}.\:\mathrm{Their} \\ $$$$\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\mathrm{joined}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fresh}\:\mathrm{lines}\:\mathrm{thus}\:\mathrm{obtained} \\ $$$$\mathrm{is} \\ $$

Question Number 21800 Answers: 0 Comments: 2

The number of integers which lie between 1 and 10^6 and which have sum of digits equal to 12 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{lie} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{10}^{\mathrm{6}} \:\mathrm{and}\:\mathrm{which}\:\mathrm{have}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{digits}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{12}\:\mathrm{is} \\ $$

Question Number 21799 Answers: 0 Comments: 2

There are n white and n red balls marked 1, 2, 3, ....n. The number of ways we can arrange these balls in a row so that neighbouring balls are of different colours is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{white}\:\mathrm{and}\:{n}\:\mathrm{red}\:\mathrm{balls} \\ $$$$\mathrm{marked}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:....{n}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{arrange}\:\mathrm{these}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neighbouring}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{of} \\ $$$$\mathrm{different}\:\mathrm{colours}\:\mathrm{is} \\ $$

Question Number 21795 Answers: 1 Comments: 3

help x∈N determine x where 7 divise 2^x +3^x

$${help} \\ $$$${x}\in{N} \\ $$$${determine}\:{x}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$$ \\ $$

Question Number 21793 Answers: 0 Comments: 6

A plank of mass 10 kg rests on a smooth horizontal surface. Two blocks A and B of masses m_A = 2 kg and m_B = 1 kg lies at a distance of 3 m on the plank. The friction coefficient between the blocks and plank are μ_A = 0.3 and μ_B = 0.1. Now a force F = 15 N is applied to the plank in horizontal direction. Find the times (in sec) after which block A collides with B.

$$\mathrm{A}\:\mathrm{plank}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{10}\:\mathrm{kg}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}.\:\mathrm{Two}\:\mathrm{blocks}\:\mathrm{A}\:\mathrm{and} \\ $$$$\mathrm{B}\:\mathrm{of}\:\mathrm{masses}\:\mathrm{m}_{\mathrm{A}} \:=\:\mathrm{2}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{m}_{\mathrm{B}} \:=\:\mathrm{1}\:\mathrm{kg} \\ $$$$\mathrm{lies}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{3}\:\mathrm{m}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plank}. \\ $$$$\mathrm{The}\:\mathrm{friction}\:\mathrm{coefficient}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{and}\:\mathrm{plank}\:\mathrm{are}\:\mu_{\mathrm{A}} \:=\:\mathrm{0}.\mathrm{3}\:\mathrm{and}\:\mu_{\mathrm{B}} \:= \\ $$$$\mathrm{0}.\mathrm{1}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{force}\:\mathrm{F}\:=\:\mathrm{15}\:\mathrm{N}\:\mathrm{is}\:\mathrm{applied}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{plank}\:\mathrm{in}\:\mathrm{horizontal}\:\mathrm{direction}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{times}\:\left(\mathrm{in}\:\mathrm{sec}\right)\:\mathrm{after}\:\mathrm{which}\:\mathrm{block}\:\mathrm{A} \\ $$$$\mathrm{collides}\:\mathrm{with}\:\mathrm{B}. \\ $$

Question Number 24687 Answers: 1 Comments: 0

A particle starts from rest at t = 0 and moves with uniform acceleration. Then (1) In any time interval starting from t = 0 the space-average of the velocity is (4/3) times of time average velocity (2) If v = v_1 at t = t_1 and v = v_2 at t = t_2 then time average velocity between t_1 and t_2 is ((v_1 + v_2 )/2) (3) Distance travelled in successive equal time intervals are in proportion of 1 : 3 : 5 ... and so on (4) If v_1 , v_2 , v_3 denote the average velocities in three successive intervals of time t_1 , t_2 , t_3 then ((v_1 − v_2 )/(v_2 − v_3 )) = ((t_1 + t_2 )/(t_2 + t_3 ))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{and} \\ $$$$\mathrm{moves}\:\mathrm{with}\:\mathrm{uniform}\:\mathrm{acceleration}.\:\mathrm{Then} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{In}\:\mathrm{any}\:\mathrm{time}\:\mathrm{interval}\:\mathrm{starting}\:\mathrm{from} \\ $$$${t}\:=\:\mathrm{0}\:\mathrm{the}\:\mathrm{space}-\mathrm{average}\:\mathrm{of}\:\mathrm{the}\:\mathrm{velocity} \\ $$$$\mathrm{is}\:\frac{\mathrm{4}}{\mathrm{3}}\:\mathrm{times}\:\mathrm{of}\:\mathrm{time}\:\mathrm{average}\:\mathrm{velocity} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{If}\:{v}\:=\:{v}_{\mathrm{1}} \:\mathrm{at}\:{t}\:=\:{t}_{\mathrm{1}} \:\mathrm{and}\:{v}\:=\:{v}_{\mathrm{2}} \:\mathrm{at}\:{t}\:=\:{t}_{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{time}\:\mathrm{average}\:\mathrm{velocity}\:\mathrm{between}\:{t}_{\mathrm{1}} \\ $$$$\mathrm{and}\:{t}_{\mathrm{2}} \:\mathrm{is}\:\frac{{v}_{\mathrm{1}} \:+\:{v}_{\mathrm{2}} }{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Distance}\:\mathrm{travelled}\:\mathrm{in}\:\mathrm{successive} \\ $$$$\mathrm{equal}\:\mathrm{time}\:\mathrm{intervals}\:\mathrm{are}\:\mathrm{in}\:\mathrm{proportion} \\ $$$$\mathrm{of}\:\mathrm{1}\::\:\mathrm{3}\::\:\mathrm{5}\:...\:\mathrm{and}\:\mathrm{so}\:\mathrm{on} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{If}\:{v}_{\mathrm{1}} ,\:{v}_{\mathrm{2}} ,\:{v}_{\mathrm{3}} \:\mathrm{denote}\:\mathrm{the}\:\mathrm{average} \\ $$$$\mathrm{velocities}\:\mathrm{in}\:\mathrm{three}\:\mathrm{successive}\:\mathrm{intervals} \\ $$$$\mathrm{of}\:\mathrm{time}\:{t}_{\mathrm{1}} ,\:{t}_{\mathrm{2}} ,\:{t}_{\mathrm{3}} \:\mathrm{then}\:\frac{{v}_{\mathrm{1}} \:−\:{v}_{\mathrm{2}} }{{v}_{\mathrm{2}} \:−\:{v}_{\mathrm{3}} }\:=\:\frac{{t}_{\mathrm{1}} \:+\:{t}_{\mathrm{2}} }{{t}_{\mathrm{2}} \:+\:{t}_{\mathrm{3}} } \\ $$

Question Number 24685 Answers: 0 Comments: 3

The sum of the kinetic and potential energies of a system of objects is conserved (1) Only when no external force acts on the objects (2) Only when the objects move along closed paths (3) Only when the work done by the resultant external force is zero (4) None of these

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{kinetic}\:\mathrm{and}\:\mathrm{potential} \\ $$$$\mathrm{energies}\:\mathrm{of}\:\mathrm{a}\:\mathrm{system}\:\mathrm{of}\:\mathrm{objects}\:\mathrm{is}\:\mathrm{conserved} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{no}\:\mathrm{external}\:\mathrm{force}\:\mathrm{acts}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{objects} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{objects}\:\mathrm{move}\:\mathrm{along} \\ $$$$\mathrm{closed}\:\mathrm{paths} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{work}\:\mathrm{done}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{resultant}\:\mathrm{external}\:\mathrm{force}\:\mathrm{is}\:\mathrm{zero} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these} \\ $$

Question Number 21772 Answers: 2 Comments: 0

integrate ∫2^(4x) dx

$${integrate} \\ $$$$\int\mathrm{2}^{\mathrm{4}{x}} {dx} \\ $$

Question Number 21771 Answers: 0 Comments: 0

The result of 11 chess matches (as win, lose or draw) are to be forecast. Out of all possible forecasts, the number of ways in which 8 correct and 3 incorrect results can be forecast is

$$\mathrm{The}\:\mathrm{result}\:\mathrm{of}\:\mathrm{11}\:\mathrm{chess}\:\mathrm{matches}\:\left(\mathrm{as}\:\mathrm{win},\right. \\ $$$$\left.\mathrm{lose}\:\mathrm{or}\:\mathrm{draw}\right)\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{forecast}.\:\mathrm{Out}\:\mathrm{of} \\ $$$$\mathrm{all}\:\mathrm{possible}\:\mathrm{forecasts},\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{8}\:\mathrm{correct}\:\mathrm{and}\:\mathrm{3}\:\mathrm{incorrect} \\ $$$$\mathrm{results}\:\mathrm{can}\:\mathrm{be}\:\mathrm{forecast}\:\mathrm{is} \\ $$

Question Number 21768 Answers: 0 Comments: 2

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

$$\mathrm{Six}\:\mathrm{cards}\:\mathrm{and}\:\mathrm{six}\:\mathrm{envelopes}\:\mathrm{are}\:\mathrm{numbered} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5},\:\mathrm{6}\:\mathrm{and}\:\mathrm{cards}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed} \\ $$$$\mathrm{in}\:\mathrm{envelopes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{each}\:\mathrm{envelope} \\ $$$$\mathrm{contains}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{card}\:\mathrm{and}\:\mathrm{no}\:\mathrm{card} \\ $$$$\mathrm{is}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{envelope}\:\mathrm{bearing}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{number}\:\mathrm{and}\:\mathrm{moreover}\:\mathrm{the}\:\mathrm{card} \\ $$$$\mathrm{numbered}\:\mathrm{1}\:\mathrm{is}\:\mathrm{always}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{envelope} \\ $$$$\mathrm{numbered}\:\mathrm{2}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\ $$$$\mathrm{it}\:\mathrm{can}\:\mathrm{be}\:\mathrm{done}\:\mathrm{is} \\ $$

Question Number 21766 Answers: 1 Comments: 0

There are 3 apartments A, B and C for rent in a building. Each apartment will accept either 3 or 4 occupants. The number of ways of renting the apartments to 10 students

$$\mathrm{There}\:\mathrm{are}\:\mathrm{3}\:\mathrm{apartments}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{for} \\ $$$$\mathrm{rent}\:\mathrm{in}\:\mathrm{a}\:\mathrm{building}.\:\mathrm{Each}\:\mathrm{apartment}\:\mathrm{will} \\ $$$$\mathrm{accept}\:\mathrm{either}\:\mathrm{3}\:\mathrm{or}\:\mathrm{4}\:\mathrm{occupants}.\:\mathrm{The} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{renting}\:\mathrm{the} \\ $$$$\mathrm{apartments}\:\mathrm{to}\:\mathrm{10}\:\mathrm{students} \\ $$

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