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Permutation and CombinationQuestion and Answers: Page 15

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

Question Number 104134 Answers: 2 Comments: 0

what is the coefficient x^(15) in the expansion of x^6 (1−x)^(11)

$${what}\:{is}\:{the}\:{coefficient}\:{x}^{\mathrm{15}} \\ $$$${in}\:{the}\:{expansion}\:{of}\:{x}^{\mathrm{6}} \left(\mathrm{1}−{x}\right)^{\mathrm{11}} \\ $$

Question Number 103903 Answers: 1 Comments: 5

a 100 cm long rod should be divided into 3 parts. the length of each part in cm should be integer. in how many different ways can this be done?

$${a}\:\mathrm{100}\:{cm}\:{long}\:{rod}\:{should}\:{be}\:{divided} \\ $$$${into}\:\mathrm{3}\:{parts}.\:{the}\:{length}\:{of}\:{each}\:{part} \\ $$$${in}\:{cm}\:{should}\:{be}\:{integer}.\:{in}\:{how}\: \\ $$$${many}\:{different}\:{ways}\:{can}\:{this}\:{be} \\ $$$${done}? \\ $$

Question Number 103826 Answers: 1 Comments: 0

In the expansion of (1+x)^(20) if the coefficient of x^r is twice the coefficient of x^(r−1) , what the value of the coefficient?

$${In}\:{the}\:{expansion}\:{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{20}} \:{if}\:{the} \\ $$$${coefficient}\:{of}\:{x}^{{r}} \:{is}\:{twice}\:{the}\:{coefficient} \\ $$$${of}\:{x}^{{r}−\mathrm{1}} ,\:{what}\:{the}\:{value}\:{of}\:{the} \\ $$$${coefficient}?\: \\ $$

Question Number 103716 Answers: 1 Comments: 3

Question Number 103606 Answers: 3 Comments: 1

an integer n between 1 and 98 , inclusive is to be chosen at random. what is the probability that n(n+1) will be divisible by 3

$${an}\:{integer}\:{n}\:{between}\:\mathrm{1}\:{and}\:\mathrm{98}\:, \\ $$$${inclusive}\:{is}\:{to}\:{be}\:{chosen}\:{at} \\ $$$${random}.\:{what}\:{is}\:{the}\:{probability} \\ $$$${that}\:{n}\left({n}+\mathrm{1}\right)\:{will}\:{be}\:{divisible}\:{by}\:\mathrm{3} \\ $$

Question Number 103513 Answers: 2 Comments: 0

coefficient of x^5 in expansion (1+x^2 )^5 (1+x)^4 equal to (a) 40 (b) 45 (c) 50 (d) 55 (e) 60

$${coefficient}\:{of}\:{x}^{\mathrm{5}} \:{in}\:{expansion} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{5}} \left(\mathrm{1}+{x}\right)^{\mathrm{4}} \:{equal}\:{to}\: \\ $$$$\left({a}\right)\:\mathrm{40}\:\:\:\:\:\:\left({b}\right)\:\mathrm{45}\:\:\:\:\:\left({c}\right)\:\mathrm{50}\:\:\:\:\left({d}\right)\:\mathrm{55} \\ $$$$\left({e}\right)\:\mathrm{60} \\ $$

Question Number 103410 Answers: 1 Comments: 0

Σ_(k=0) ^n (((−1)^k )/(k!))=?

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}=? \\ $$

Question Number 103406 Answers: 0 Comments: 3

100 students are standing in a row. 4 students should be selected in the way that no two of them are next to each other. how many ways do you have to do this?

$$\mathrm{100}\:{students}\:{are}\:{standing}\:{in}\:{a}\:{row}. \\ $$$$\mathrm{4}\:{students}\:{should}\:{be}\:{selected}\:{in}\:{the} \\ $$$${way}\:{that}\:{no}\:{two}\:{of}\:{them}\:{are}\:{next}\:{to} \\ $$$${each}\:{other}.\:{how}\:{many}\:{ways}\:{do}\:{you} \\ $$$${have}\:{to}\:{do}\:{this}? \\ $$

Question Number 103367 Answers: 1 Comments: 0

many words with 4 letters can be formed using letters from the word TINKUTARA is ___

$${many}\:{words}\:{with}\:\mathrm{4}\:{letters}\:{can}\:{be}\:{formed} \\ $$$${using}\:{letters}\:{from}\:{the}\:{word} \\ $$$${TINKUTARA}\:{is}\:\_\_\_ \\ $$

Question Number 103357 Answers: 1 Comments: 0

There are 21 students will be trained with 6 trainers available. Every students is trained by 1 coach and every coach trains students with different amounts. many ways of grouping students who will be trained are .... (a) ((21!)/(6!)) (b) 6!.21! (c) ((21!)/(2!.3!.4!.5! )) (d) 6×21! (e) ((21!)/(15!))×6!

$${There}\:{are}\:\mathrm{21}\:{students}\:{will}\:{be}\:{trained} \\ $$$${with}\:\mathrm{6}\:{trainers}\:{available}.\:{Every}\:{students} \\ $$$${is}\:{trained}\:{by}\:\mathrm{1}\:{coach}\:{and}\:{every}\:{coach} \\ $$$${trains}\:{students}\:{with}\:{different}\:{amounts}. \\ $$$${many}\:{ways}\:{of}\:{grouping}\:{students}\:{who} \\ $$$${will}\:{be}\:{trained}\:{are}\:.... \\ $$$$\left({a}\right)\:\frac{\mathrm{21}!}{\mathrm{6}!}\:\:\:\:\left({b}\right)\:\mathrm{6}!.\mathrm{21}!\:\:\:\:\:\left({c}\right)\:\frac{\mathrm{21}!}{\mathrm{2}!.\mathrm{3}!.\mathrm{4}!.\mathrm{5}!\:\:}\:\: \\ $$$$\left({d}\right)\:\mathrm{6}×\mathrm{21}!\:\:\:\:\left({e}\right)\:\frac{\mathrm{21}!}{\mathrm{15}!}×\mathrm{6}! \\ $$

Question Number 103300 Answers: 1 Comments: 0

many positive five−digit integers with the first number 1 and there are three equal numbers ? (a) 810 (b) 720 (c)120 (d) 60 (e) 20

$$\mathrm{many}\:\mathrm{positive}\:\mathrm{five}−\mathrm{digit} \\ $$$$\mathrm{integers}\:\mathrm{with}\:\mathrm{the}\:\mathrm{first}\:\mathrm{number}\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{there}\:\mathrm{are}\:\mathrm{three}\:\mathrm{equal} \\ $$$$\mathrm{numbers}\:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{810}\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{720}\:\:\:\:\left(\mathrm{c}\right)\mathrm{120} \\ $$$$\left(\mathrm{d}\right)\:\mathrm{60}\:\:\:\:\:\left(\mathrm{e}\right)\:\mathrm{20} \\ $$

Question Number 103294 Answers: 1 Comments: 0

from letters in ′MATEMATIKA′ words formed by using all the letters . How many words that can be formed with the five consonant are always side by side

$$\mathrm{from}\:\mathrm{letters}\:\mathrm{in} \\ $$$$'\mathrm{MATEMATIKA}'\:\mathrm{words} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{using}\:\mathrm{all}\:\mathrm{the}\:\mathrm{letters}\:. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{words}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{with}\:\mathrm{the}\:\mathrm{five}\:\mathrm{consonant} \\ $$$$\mathrm{are}\:\mathrm{always}\:\mathrm{side}\:\mathrm{by}\:\mathrm{side}\: \\ $$

Question Number 102816 Answers: 2 Comments: 28

How many 6 digit numbers exist whose digits have exactly the sum 13? for example 120505 is such a number.

$${How}\:{many}\:\mathrm{6}\:{digit}\:{numbers}\:{exist} \\ $$$${whose}\:{digits}\:{have}\:{exactly}\:{the}\:{sum}\:\mathrm{13}? \\ $$$$ \\ $$$${for}\:{example}\:\mathrm{120505}\:{is}\:{such}\:{a}\:{number}. \\ $$

Question Number 102800 Answers: 1 Comments: 0

There are 14 boys and 10 girls in a classroom. The teacher wants to form a team of 5 students . The team must have a least two boys and two girls . Find the number of ways the team can be chosen.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{14}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{10}\:\mathrm{girls}\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{classroom}.\:\mathrm{The}\:\mathrm{teacher}\:\mathrm{wants} \\ $$$$\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{team}\:\mathrm{of}\:\mathrm{5}\:\mathrm{students}\:. \\ $$$$\mathrm{The}\:\mathrm{team}\:\mathrm{must}\:\mathrm{have}\:\mathrm{a}\:\mathrm{least}\:\mathrm{two} \\ $$$$\mathrm{boys}\:\mathrm{and}\:\mathrm{two}\:\mathrm{girls}\:.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{the}\:\mathrm{team}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{chosen}.\: \\ $$

Question Number 102329 Answers: 0 Comments: 0

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