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Question Number 21931 Answers: 0 Comments: 2

The number of ways of distributing six identical mathematics books and six identical physics books among three students such that each student gets atleast one mathematics book and atleast one physics book is ((5.5!)/k), then k is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{distributing}\:\mathrm{six} \\ $$$$\mathrm{identical}\:\mathrm{mathematics}\:\mathrm{books}\:\mathrm{and}\:\mathrm{six} \\ $$$$\mathrm{identical}\:\mathrm{physics}\:\mathrm{books}\:\mathrm{among}\:\mathrm{three} \\ $$$$\mathrm{students}\:\mathrm{such}\:\mathrm{that}\:\mathrm{each}\:\mathrm{student}\:\mathrm{gets} \\ $$$$\mathrm{atleast}\:\mathrm{one}\:\mathrm{mathematics}\:\mathrm{book}\:\mathrm{and} \\ $$$$\mathrm{atleast}\:\mathrm{one}\:\mathrm{physics}\:\mathrm{book}\:\mathrm{is}\:\frac{\mathrm{5}.\mathrm{5}!}{{k}},\:\mathrm{then}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21930 Answers: 0 Comments: 3

An eight digit number is formed from 1, 2, 3, 4 such that product of all digits is always 3072, the total number of ways is (23.^8 C_k ), where the value of k is

$$\mathrm{An}\:\mathrm{eight}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{is}\:\mathrm{formed}\:\mathrm{from} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\mathrm{such}\:\mathrm{that}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits} \\ $$$$\mathrm{is}\:\mathrm{always}\:\mathrm{3072},\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{is}\:\left(\mathrm{23}.\:^{\mathrm{8}} {C}_{{k}} \right),\:\mathrm{where}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21929 Answers: 0 Comments: 2

There are 8 Hindi novels and 6 English novels. 4 Hindi novels and 3 English novels are selected and arranged in a row such that they are alternate then no. of ways is

$$\mathrm{There}\:\mathrm{are}\:\mathrm{8}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{6}\:\mathrm{English} \\ $$$$\mathrm{novels}.\:\mathrm{4}\:\mathrm{Hindi}\:\mathrm{novels}\:\mathrm{and}\:\mathrm{3}\:\mathrm{English} \\ $$$$\mathrm{novels}\:\mathrm{are}\:\mathrm{selected}\:\mathrm{and}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{such}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{alternate}\:\mathrm{then} \\ $$$$\mathrm{no}.\:\mathrm{of}\:\mathrm{ways}\:\mathrm{is} \\ $$

Question Number 21925 Answers: 1 Comments: 0

In an xy plane the graph of the equation (x−6)^2 +(y+5)^2 =16 is a circle. P(10,−5) is on the circle. If PQ is a diameter of the circle, what is the co-ordinate at Q?

$$\mathrm{In}\:\mathrm{an}\:\mathrm{xy}\:\mathrm{plane}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left(\mathrm{x}−\mathrm{6}\right)^{\mathrm{2}} +\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} =\mathrm{16}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}.\:\mathrm{P}\left(\mathrm{10},−\mathrm{5}\right) \\ $$$$\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}.\:\mathrm{If}\:\mathrm{PQ}\:\mathrm{is}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{circle},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{co}-\mathrm{ordinate}\:\mathrm{at}\:\mathrm{Q}? \\ $$

Question Number 21917 Answers: 0 Comments: 4

How many seven letter words can be formed by using the letters of the word SUCCESS so that neither two C nor two S are together?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{seven}\:\mathrm{letter}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{using}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{SUCCESS}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neither}\:\mathrm{two}\:\mathrm{C}\:\mathrm{nor} \\ $$$$\mathrm{two}\:\mathrm{S}\:\mathrm{are}\:\mathrm{together}? \\ $$

Question Number 21913 Answers: 0 Comments: 0

Let a_n denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_n = the number of such n-digit integers ending with digit 1 and c_n = the number of such n-digit integers ending with digit 0. 1. Which of the following is correct? (1) a_(17) = a_(16) + a_(15) (2) c_(17) ≠ c_(16) + c_(15) (3) b_(17) ≠ b_(16) + c_(16) (4) a_(17) = c_(17) + b_(16) 2. The value of b_6 is

$$\mathrm{Let}\:{a}_{{n}} \:\mathrm{denote}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{all}\:{n}-\mathrm{digit} \\ $$$$\mathrm{positive}\:\mathrm{integers}\:\mathrm{formed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{0},\:\mathrm{1}\:\mathrm{or}\:\mathrm{both}\:\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{in}\:\mathrm{them}\:\mathrm{are}\:\mathrm{0}.\:\mathrm{Let}\:{b}_{{n}} \:=\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{such}\:{n}-\mathrm{digit}\:\mathrm{integers}\:\mathrm{ending} \\ $$$$\mathrm{with}\:\mathrm{digit}\:\mathrm{1}\:\mathrm{and}\:{c}_{{n}} \:=\:\mathrm{the}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{such}\:{n}-\mathrm{digit}\:\mathrm{integers}\:\mathrm{ending}\:\mathrm{with} \\ $$$$\mathrm{digit}\:\mathrm{0}. \\ $$$$\mathrm{1}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{correct}? \\ $$$$\left(\mathrm{1}\right)\:{a}_{\mathrm{17}} \:=\:{a}_{\mathrm{16}} \:+\:{a}_{\mathrm{15}} \\ $$$$\left(\mathrm{2}\right)\:{c}_{\mathrm{17}} \:\neq\:{c}_{\mathrm{16}} \:+\:{c}_{\mathrm{15}} \\ $$$$\left(\mathrm{3}\right)\:{b}_{\mathrm{17}} \:\neq\:{b}_{\mathrm{16}} \:+\:{c}_{\mathrm{16}} \\ $$$$\left(\mathrm{4}\right)\:{a}_{\mathrm{17}} \:=\:{c}_{\mathrm{17}} \:+\:{b}_{\mathrm{16}} \\ $$$$\mathrm{2}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{b}_{\mathrm{6}} \:\mathrm{is} \\ $$

Question Number 21904 Answers: 0 Comments: 3

Compute the area of a loop of the curve ((x^2 /a^2 )+(y^2 /b^2 ))=((2xy)/c^2 ) .

$${Compute}\:{the}\:{area}\:{of}\:{a}\:{loop}\:{of}\:{the} \\ $$$${curve}\:\left(\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\right)=\frac{\mathrm{2}{xy}}{{c}^{\mathrm{2}} }\:. \\ $$

Question Number 21903 Answers: 0 Comments: 0

ΔABC with sides a, b, c ∈ Z and ∠A = 3∠B If its circumference is minimum, then find a, b, c

$$\Delta{ABC}\:\mathrm{with}\:\mathrm{sides}\:{a},\:{b},\:{c}\:\in\:\mathbb{Z}\:\mathrm{and}\:\angle{A}\:=\:\mathrm{3}\angle{B} \\ $$$$\mathrm{If}\:\mathrm{its}\:\mathrm{circumference}\:\mathrm{is}\:\mathrm{minimum},\:\mathrm{then}\:\mathrm{find}\:{a},\:{b},\:{c} \\ $$

Question Number 21889 Answers: 1 Comments: 0

If a,b, and A of a triangle are fixed and two possible values of the third side be c_1 and c_2 such that c_1 ^2 +c_1 c_2 +c_2 ^2 =a^2 , then find angle A.

$${If}\:{a},{b},\:{and}\:{A}\:{of}\:{a}\:{triangle}\:\:{are}\: \\ $$$${fixed}\:{and}\:{two}\:{possible}\:{values}\:{of}\:{the}\: \\ $$$${third}\:{side}\:{be}\:{c}_{\mathrm{1}} {and}\:{c}_{\mathrm{2}} {such}\:{that} \\ $$$$\boldsymbol{{c}}_{\mathrm{1}} ^{\mathrm{2}} +\boldsymbol{{c}}_{\mathrm{1}} \boldsymbol{{c}}_{\mathrm{2}} +\boldsymbol{{c}}_{\mathrm{2}} ^{\mathrm{2}} =\boldsymbol{{a}}^{\mathrm{2}} ,\:{then}\:{find}\:{angle}\:{A}. \\ $$

Question Number 21877 Answers: 2 Comments: 3

x=^3 (√(7+5(√2)))+^3 (√(7−5(√2))) 1. According to a video, x=2 2. According to WolframAlpha, x≈0.2071+0.3587i for “principal root” and x=2(√2) for “real-valued root” 3. According to google, x≈2.8284 Please help and explain! :)

$${x}=^{\mathrm{3}} \sqrt{\mathrm{7}+\mathrm{5}\sqrt{\mathrm{2}}}+^{\mathrm{3}} \sqrt{\mathrm{7}−\mathrm{5}\sqrt{\mathrm{2}}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{a}\:\mathrm{video},\:{x}=\mathrm{2} \\ $$$$\: \\ $$$$\mathrm{2}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{WolframAlpha}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{x}\approx\mathrm{0}.\mathrm{2071}+\mathrm{0}.\mathrm{3587}{i}\:\:\mathrm{for}\:``\mathrm{principal}\:\mathrm{root}'' \\ $$$$\mathrm{and}\:\:\:{x}=\mathrm{2}\sqrt{\mathrm{2}}\:\:\mathrm{for}\:``\mathrm{real}-\mathrm{valued}\:\mathrm{root}'' \\ $$$$\: \\ $$$$\mathrm{3}.\:\mathrm{According}\:\mathrm{to}\:\mathrm{google},\:{x}\approx\mathrm{2}.\mathrm{8284} \\ $$$$\: \\ $$$$\left.\mathrm{Please}\:\mathrm{help}\:\mathrm{and}\:\mathrm{explain}!\:\:\::\right) \\ $$

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

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

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