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Permutation and CombinationQuestion and Answers: Page 13

Question Number 111394 Answers: 1 Comments: 2

A teacher conducts a test for five students. He provides the marking scheme and asked them to exchange their scripts such that none of them marks his own script. How many ways can the students carry out the marking?

$$\mathrm{A}\:\mathrm{teacher}\:\mathrm{conducts}\:\mathrm{a}\:\mathrm{test}\:\mathrm{for}\:\mathrm{five} \\ $$$$\mathrm{students}.\:\mathrm{He}\:\mathrm{provides}\:\mathrm{the}\:\mathrm{marking} \\ $$$$\mathrm{scheme}\:\mathrm{and}\:\mathrm{asked}\:\mathrm{them}\:\mathrm{to}\:\mathrm{exchange} \\ $$$$\mathrm{their}\:\mathrm{scripts}\:\mathrm{such}\:\mathrm{that}\:\mathrm{none}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{marks}\:\mathrm{his}\:\mathrm{own}\:\mathrm{script}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways} \\ $$$$\mathrm{can}\:\mathrm{the}\:\mathrm{students}\:\mathrm{carry}\:\mathrm{out}\:\mathrm{the} \\ $$$$\mathrm{marking}? \\ $$

Question Number 111272 Answers: 1 Comments: 0

Tricolours flags(each flag having three different strips of non−overlapping colours) are to be designed using white,blue,red,yellow and black strips. How many of the flags have blue colour?

$$\mathrm{Tricolours}\:\mathrm{flags}\left(\mathrm{each}\:\mathrm{flag}\:\mathrm{having}\right. \\ $$$$\mathrm{three}\:\mathrm{different}\:\mathrm{strips}\:\mathrm{of} \\ $$$$\left.\mathrm{non}−\mathrm{overlapping}\:\mathrm{colours}\right)\:\mathrm{are}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{designed}\:\mathrm{using}\:\mathrm{white},\mathrm{blue},\mathrm{red},\mathrm{yellow} \\ $$$$\mathrm{and}\:\mathrm{black}\:\mathrm{strips}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{flags}\:\mathrm{have}\:\mathrm{blue}\:\mathrm{colour}? \\ $$

Question Number 111275 Answers: 1 Comments: 0

In a tennis tournament n women and 2n men played. Each player played exactly one match with every other player. If there are no ties and the number of the matches won by women to the number of matches won by men is 7:5, find n.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{tennis}\:\mathrm{tournament}\:\mathrm{n}\:\mathrm{women}\:\mathrm{and} \\ $$$$\mathrm{2n}\:\mathrm{men}\:\mathrm{played}.\:\mathrm{Each}\:\mathrm{player}\:\mathrm{played} \\ $$$$\mathrm{exactly}\:\mathrm{one}\:\mathrm{match}\:\mathrm{with}\:\mathrm{every}\:\mathrm{other} \\ $$$$\mathrm{player}.\:\mathrm{If}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{ties}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{women} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{matches}\:\mathrm{won}\:\mathrm{by}\:\mathrm{men} \\ $$$$\mathrm{is}\:\mathrm{7}:\mathrm{5},\:\mathrm{find}\:\mathrm{n}. \\ $$$$ \\ $$

Question Number 110729 Answers: 1 Comments: 0

How many ways can the letters in the word MATHEMATICS be rearranged such that the word formed neither starts nor ends with a vowel, and any four consecutive letters must contain at least a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{word}\:\boldsymbol{\mathrm{MATHEMATICS}} \\ $$$$\mathrm{be}\:\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{formed}\:\mathrm{neither}\:\mathrm{starts}\:\mathrm{nor}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\: \\ $$$$\mathrm{vowel},\:\mathrm{and}\:\mathrm{any}\:\mathrm{four}\:\mathrm{consecutive}\: \\ $$$$\mathrm{letters}\:\mathrm{must}\:\mathrm{contain}\:\mathrm{at}\:\mathrm{least}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Question Number 110524 Answers: 1 Comments: 0

A blind man is to place 6 letters into 6 pigeon holes, how many ways can atleast 5 letters be wrongly placed? (Note that only one letter must be in a pigeon hole).

$$\mathrm{A}\:\mathrm{blind}\:\mathrm{man}\:\mathrm{is}\:\mathrm{to}\:\mathrm{place}\:\mathrm{6}\:\mathrm{letters}\:\mathrm{into}\:\mathrm{6} \\ $$$$\mathrm{pigeon}\:\mathrm{holes},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can} \\ $$$$\mathrm{atleast}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{be}\:\mathrm{wrongly}\:\mathrm{placed}? \\ $$$$\left(\mathrm{Note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{pigeon}\:\mathrm{hole}\right). \\ $$

Question Number 110523 Answers: 0 Comments: 0

In the square PQRS, K is the midpoint of PQ, L is the midpoint of QR, M is the midpoint RS, N is the midpoint of SP and O is the midpoint of KM. A line segment is drawn from each pair of points from (K,L,M,N,O,P,Q,R,S). These line segments create points of intersections not contained in (K,L,M,N,O,P,Q,R,S). How many distinct such points are there?

$$ \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{square}\:\mathrm{PQRS},\:\mathrm{K}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{PQ},\:\mathrm{L}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{QR},\:\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{RS},\:\mathrm{N}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{SP}\:\mathrm{and}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$$\mathrm{of}\:\mathrm{KM}.\:\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{points}\:\mathrm{from} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{These}\:\mathrm{line} \\ $$$$\mathrm{segments}\:\mathrm{create}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{not}\:\mathrm{contained}\:\mathrm{in} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{distinct}\:\mathrm{such}\:\mathrm{points}\:\mathrm{are}\:\mathrm{there}? \\ $$

Question Number 110498 Answers: 1 Comments: 3

How many ways can the letters in the word MATHEMATICS be rearranged such that the word formed either starts or ends with a vowel, and any three consecutive letters must contain a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{word}\:\mathrm{MATHEMATICS}\:\mathrm{be} \\ $$$$\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word}\:\mathrm{formed} \\ $$$$\mathrm{either}\:\mathrm{starts}\:\mathrm{or}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\:\mathrm{vowel},\:\mathrm{and} \\ $$$$\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{letters}\:\mathrm{must} \\ $$$$\mathrm{contain}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Question Number 109488 Answers: 0 Comments: 0

How many are the permutations of 1 − a little rubik′s cube with 4 squares by side 2 − a classical one with 9 squares by side

$${How}\:{many}\:{are}\:{the}\:{permutations}\:{of} \\ $$$$\mathrm{1}\:−\:{a}\:{little}\:{rubik}'{s}\:{cube}\:{with}\:\mathrm{4}\:{squares}\:{by}\:{side} \\ $$$$\mathrm{2}\:−\:{a}\:{classical}\:{one}\:{with}\:\mathrm{9}\:{squares}\:{by}\:{side} \\ $$

Question Number 109387 Answers: 0 Comments: 1

SUCCESSFULLY How many different words can you form using these letters so that no two same letters are adjacent?

$$\boldsymbol{\mathrm{SUCCESSFULLY}} \\ $$$${How}\:{many}\:{different}\:{words}\:{can}\:{you} \\ $$$${form}\:{using}\:{these}\:{letters}\:{so}\:{that}\:{no} \\ $$$${two}\:{same}\:{letters}\:{are}\:{adjacent}? \\ $$

Question Number 108638 Answers: 1 Comments: 0

((bemath)/★) prove that (((n−1)),(( r)) ) + (((n−1)),((r−1)) ) = ((n),(r) )

$$\:\:\:\frac{{bemath}}{\bigstar} \\ $$$${prove}\:{that}\:\begin{pmatrix}{{n}−\mathrm{1}}\\{\:\:\:\:\:{r}}\end{pmatrix}\:+\:\begin{pmatrix}{{n}−\mathrm{1}}\\{{r}−\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix} \\ $$

Question Number 108450 Answers: 2 Comments: 2

((BeMath)/(⊂⊃)) (1)find ((1/2))! (2)∫_0 ^(π/2) ((x sin x)/((1+cos x)^2 )) dx

$$\:\:\frac{\mathcal{B}{e}\mathcal{M}{ath}}{\subset\supset} \\ $$$$\left(\mathrm{1}\right){find}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)! \\ $$$$\left(\mathrm{2}\right)\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{{x}\:\mathrm{sin}\:{x}}{\left(\mathrm{1}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 108262 Answers: 2 Comments: 0

∫_0 ^1 (y^y )^((y^y )^((y^y )) ) dy=? please help

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{y}^{\mathrm{y}} \right)^{\left(\mathrm{y}^{\mathrm{y}} \right)^{\left(\mathrm{y}^{\mathrm{y}} \right)} } \mathrm{dy}=? \\ $$$$\mathrm{please}\:\mathrm{help} \\ $$

Question Number 107844 Answers: 0 Comments: 0

A general case: we have totally n letters, among them n_1 times A, n_2 times B, n_3 times C, n_4 times D etc. (n_1 ,n_2 ,n_3 ,n_(4,) ...≥2, n>n_1 +n_2 +n_3 +n_4 +....) how many different words can be formed using these n letters such that same letters are not next to each other. see also Q107451.

$${A}\:{general}\:{case}: \\ $$$${we}\:{have}\:{totally}\:{n}\:{letters},\:{among}\:{them} \\ $$$${n}_{\mathrm{1}} \:{times}\:{A},\:{n}_{\mathrm{2}} \:{times}\:{B},\:{n}_{\mathrm{3}} \:{times}\:{C}, \\ $$$${n}_{\mathrm{4}} \:{times}\:{D}\:{etc}. \\ $$$$\left({n}_{\mathrm{1}} ,{n}_{\mathrm{2}} ,{n}_{\mathrm{3}} ,{n}_{\mathrm{4},} ...\geqslant\mathrm{2},\:{n}>{n}_{\mathrm{1}} +{n}_{\mathrm{2}} +{n}_{\mathrm{3}} +{n}_{\mathrm{4}} +....\right) \\ $$$${how}\:{many}\:{different}\:{words}\:{can}\:{be} \\ $$$${formed}\:{using}\:{these}\:{n}\:{letters}\:{such}\:{that} \\ $$$${same}\:{letters}\:{are}\:{not}\:{next}\:{to}\:{each} \\ $$$${other}. \\ $$$$ \\ $$$${see}\:{also}\:{Q}\mathrm{107451}. \\ $$

Question Number 107451 Answers: 1 Comments: 1

How many words can you form using the letters in UNUSUALLY such that no same letters are next to each other? [Answer: 10200]

$${How}\:{many}\:{words}\:{can}\:{you}\:{form}\:{using} \\ $$$${the}\:{letters}\:\:{in}\:\boldsymbol{{UNUSUALLY}} \\ $$$${such}\:{that}\:{no}\:{same}\:{letters}\:{are}\:\:{next} \\ $$$${to}\:{each}\:{other}? \\ $$$$ \\ $$$$\left[{Answer}:\:\mathrm{10200}\right] \\ $$

Question Number 107327 Answers: 0 Comments: 0

1≤p≤k≤n show that Σ_(k=p) ^n C_(k−1) ^(p−1) =C_n ^p please i need help

$$\mathrm{1}\leqslant{p}\leqslant{k}\leqslant{n} \\ $$$${show}\:{that}\:\underset{{k}={p}} {\overset{{n}} {\sum}}\boldsymbol{{C}}_{{k}−\mathrm{1}} ^{{p}−\mathrm{1}} =\boldsymbol{{C}}_{{n}} ^{{p}} \\ $$$${please}\:{i}\:{need}\:{help} \\ $$

Question Number 106815 Answers: 1 Comments: 0

please help me to show that the equation X^n +aX+c=0 can not have more than 3 reals solutions

$$\:\:\:\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{to}}\:\boldsymbol{{show}} \\ $$$$\boldsymbol{{that}}\:\:\boldsymbol{{the}}\:\boldsymbol{{equation}}\: \\ $$$$\:\boldsymbol{{X}}^{\boldsymbol{{n}}} +\boldsymbol{{aX}}+\boldsymbol{{c}}=\mathrm{0}\:\boldsymbol{{can}}\:\boldsymbol{{not}}\:\boldsymbol{{have}} \\ $$$$\boldsymbol{{more}}\:\boldsymbol{{than}}\:\mathrm{3}\:\boldsymbol{{reals}}\:\boldsymbol{{solutions}} \\ $$$$ \\ $$

Question Number 106792 Answers: 1 Comments: 0

≻bobhans≺ From a batch containing 6 boys and 4 girls a group of 4 students is tobe selected . How many group formations will have exactly 2 girls?

$$\:\:\:\:\:\:\succ\mathrm{bobhans}\prec \\ $$$$\mathrm{From}\:\mathrm{a}\:\mathrm{batch}\:\mathrm{containing}\:\mathrm{6}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{4}\:\mathrm{girls} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{4}\:\mathrm{students}\:\mathrm{is}\:\mathrm{tobe}\:\mathrm{selected}\:. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{group}\:\mathrm{formations}\:\mathrm{will}\:\mathrm{have} \\ $$$$\mathrm{exactly}\:\mathrm{2}\:\mathrm{girls}? \\ $$

Question Number 106305 Answers: 1 Comments: 0

what is probability of 5 coming up at least one if a die is rolled 3 times

$$\mathrm{what}\:\mathrm{is}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{5}\:\mathrm{coming}\:\mathrm{up}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{if}\:\mathrm{a}\:\mathrm{die}\: \\ $$$$\mathrm{is}\:\mathrm{rolled}\:\mathrm{3}\:\mathrm{times}\: \\ $$

Question Number 106292 Answers: 1 Comments: 0

a box contains 4 blue, 3 green and 2 red identicall balls. if two balls are selected at random without replacement , what is the probability that two balls be of the same colours?

$$\mathrm{a}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{4}\:\mathrm{blue},\:\mathrm{3}\:\mathrm{green}\:\mathrm{and}\:\mathrm{2}\:\mathrm{red}\:\mathrm{identicall}\:\mathrm{balls}.\: \\ $$$$\mathrm{if}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{without}\: \\ $$$$\mathrm{replacement}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\mathrm{that}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{be}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{colours}? \\ $$

Question Number 105994 Answers: 1 Comments: 1

Given { ((2log _5 (x+2)−log _5 (y)+(1/2)=0)),((log _5 (x^2 +2x−2)=0 )) :} find the value of y

$$\mathbb{G}{iven}\:\begin{cases}{\mathrm{2log}\:_{\mathrm{5}} \left({x}+\mathrm{2}\right)−\mathrm{log}\:_{\mathrm{5}} \left({y}\right)+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}}\\{\mathrm{log}\:_{\mathrm{5}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{2}\right)=\mathrm{0}\:}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:{y}\: \\ $$

Question Number 105853 Answers: 0 Comments: 5

A box contains 5 balls, 2 balls were drawn at random, both of which turned out tobe white. what is the probability that all the balls in the box are white ?

$${A}\:{box}\:{contains}\:\mathrm{5}\:{balls},\:\mathrm{2}\:{balls}\:{were} \\ $$$${drawn}\:{at}\:{random},\:{both}\:{of}\:{which} \\ $$$${turned}\:{out}\:{tobe}\:{white}.\:{what}\:{is}\:{the} \\ $$$${probability}\:{that}\:{all}\:{the}\:{balls}\:{in}\:{the}\:{box} \\ $$$${are}\:{white}\:? \\ $$

Question Number 105791 Answers: 1 Comments: 1

From 1 to 12345, how many numbers contain the digit 0? Find the number of zeros in all these numbers. Example: 10020 has three zeros.

$${From}\:\mathrm{1}\:{to}\:\mathrm{12345},\:{how}\:{many}\:{numbers} \\ $$$${contain}\:{the}\:{digit}\:\mathrm{0}?\:{Find}\:{the}\:{number} \\ $$$${of}\:{zeros}\:{in}\:{all}\:{these}\:{numbers}. \\ $$$${Example}:\:\mathrm{10020}\:{has}\:{three}\:{zeros}. \\ $$

Question Number 105551 Answers: 2 Comments: 0

In how many different ways can 10 students be divided into 3 groups?

$${In}\:{how}\:{many}\:{different}\:{ways}\:{can}\:\mathrm{10} \\ $$$${students}\:{be}\:{divided}\:{into}\:\mathrm{3}\:{groups}? \\ $$

Question Number 105265 Answers: 3 Comments: 0

Question Number 104370 Answers: 1 Comments: 0

Question Number 104155 Answers: 0 Comments: 3

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