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Permutation and CombinationQuestion and Answers: Page 24

Question Number 21683 Answers: 1 Comments: 2

Suppose A_1 A_2 ...A_(20) is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are not the sides of the polygon?

$$\mathrm{Suppose}\:{A}_{\mathrm{1}} {A}_{\mathrm{2}} ...{A}_{\mathrm{20}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{20}-\mathrm{sided}\:\mathrm{regular} \\ $$$$\mathrm{polygon}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{non}-\mathrm{isosceles} \\ $$$$\left(\mathrm{scalene}\right)\:\mathrm{triangles}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{whose} \\ $$$$\mathrm{vertices}\:\mathrm{are}\:\mathrm{among}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{polygon}\:\mathrm{but}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}? \\ $$

Question Number 21651 Answers: 0 Comments: 4

How many four-digit numbers are there whose decimal notation contains not more than two distinct digits?

$${How}\:{many}\:{four}-{digit}\:{numbers}\:{are} \\ $$$${there}\:{whose}\:{decimal}\:{notation}\:{contains} \\ $$$${not}\:{more}\:{than}\:{two}\:{distinct}\:{digits}? \\ $$

Question Number 21693 Answers: 0 Comments: 15

How many six-digit numbers contain exactly four different digits?

$${How}\:{many}\:{six}-{digit}\:{numbers}\:{contain} \\ $$$${exactly}\:{four}\:{different}\:{digits}? \\ $$

Question Number 21598 Answers: 0 Comments: 0

Prove that ((n^2 !)/((n!)^n )) is an integer, n ∈ N.

$${Prove}\:{that}\:\frac{{n}^{\mathrm{2}} !}{\left({n}!\right)^{{n}} }\:{is}\:{an}\:{integer},\:{n}\:\in\:{N}. \\ $$

Question Number 21587 Answers: 1 Comments: 1

If n objects are arranged in a row, then find the number of ways of selecting three of these objects so that no two of them are next to each other.

$$\mathrm{If}\:{n}\:\mathrm{objects}\:\mathrm{are}\:\mathrm{arranged}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{selecting} \\ $$$$\mathrm{three}\:\mathrm{of}\:\mathrm{these}\:\mathrm{objects}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of} \\ $$$$\mathrm{them}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$

Question Number 21572 Answers: 0 Comments: 0

Determine the largest 3-digit prime factor of the integer^(2000) C_(1000) .

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{3}-\mathrm{digit}\:\mathrm{prime} \\ $$$$\mathrm{factor}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integer}\:^{\mathrm{2000}} {C}_{\mathrm{1000}} . \\ $$

Question Number 21406 Answers: 0 Comments: 2

Find the number of ways in which n distinct balls can be put into three boxes so that no two boxes remain empty.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:{n} \\ $$$$\mathrm{distinct}\:\mathrm{balls}\:\mathrm{can}\:\mathrm{be}\:\mathrm{put}\:\mathrm{into}\:\mathrm{three} \\ $$$$\mathrm{boxes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{boxes}\:\mathrm{remain} \\ $$$$\mathrm{empty}. \\ $$

Question Number 21405 Answers: 1 Comments: 0

Four dice are rolled. The number of ways in which at least one die shows 3, is

$$\mathrm{Four}\:\mathrm{dice}\:\mathrm{are}\:\mathrm{rolled}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{die}\:\mathrm{shows}\:\mathrm{3},\:\mathrm{is} \\ $$

Question Number 21404 Answers: 1 Comments: 0

Prove that (6n)! is divisible by 2^(2n) .3^n .

$$\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{6}{n}\right)!\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}^{\mathrm{2}{n}} .\mathrm{3}^{{n}} . \\ $$

Question Number 21374 Answers: 0 Comments: 2

In how many ways can the letters of the word PATLIPUTRA be arranged, so that the relative order of vowels are consonants do not alter?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{word}\:\mathrm{PATLIPUTRA}\:\mathrm{be}\:\mathrm{arranged},\:\mathrm{so} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{relative}\:\mathrm{order}\:\mathrm{of}\:\mathrm{vowels}\:\mathrm{are} \\ $$$$\mathrm{consonants}\:\mathrm{do}\:\mathrm{not}\:\mathrm{alter}? \\ $$

Question Number 20926 Answers: 0 Comments: 0

Question Number 17770 Answers: 1 Comments: 0

Find the number of ways the digits 0,1,2 and 3 can be permuted to give rise to a number greater than 2000.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{the} \\ $$$$\mathrm{digits}\:\mathrm{0},\mathrm{1},\mathrm{2}\:\mathrm{and}\:\mathrm{3}\:\mathrm{can}\:\mathrm{be}\:\mathrm{permuted} \\ $$$$\mathrm{to}\:\mathrm{give}\:\mathrm{rise}\:\mathrm{to}\:\mathrm{a}\:\mathrm{number}\:\mathrm{greater} \\ $$$$\mathrm{than}\:\mathrm{2000}. \\ $$

Question Number 16876 Answers: 1 Comments: 0

In how many ways can a family of 5 brothers be seated round a table if (i) 2 brothers must seat next to each other. (ii) 2 brothers must not seat together.

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{a}\:\mathrm{family}\:\mathrm{of}\:\mathrm{5}\:\mathrm{brothers}\:\mathrm{be}\:\mathrm{seated}\:\mathrm{round}\:\mathrm{a}\:\mathrm{table} \\ $$$$\mathrm{if}\:\left(\mathrm{i}\right)\:\mathrm{2}\:\mathrm{brothers}\:\mathrm{must}\:\mathrm{seat}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{2}\:\mathrm{brothers}\:\mathrm{must}\:\mathrm{not}\:\mathrm{seat}\:\mathrm{together}. \\ $$

Question Number 16868 Answers: 1 Comments: 0

In how many ways can the letters of the word. EVERMORE be arrange if the word must begin with (i) R (ii) E

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}.\:\mathrm{EVERMORE}\:\mathrm{be}\:\mathrm{arrange} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{word}\:\mathrm{must}\:\mathrm{begin}\:\mathrm{with}\: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{R} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{E} \\ $$

Question Number 16836 Answers: 1 Comments: 0

Find how many number greater than 2,500 can be formed from the digit 0, 1, 2, 3, 4 if no digit can be used more than once.

$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{number}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2},\mathrm{500}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digit} \\ $$$$\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\:\mathrm{if}\:\mathrm{no}\:\mathrm{digit}\:\mathrm{can}\:\mathrm{be}\:\mathrm{used}\:\mathrm{more}\:\mathrm{than}\:\mathrm{once}. \\ $$

Question Number 13508 Answers: 1 Comments: 0

5!! = ? please workings, how is the answer 15

$$\mathrm{5}!!\:=\:? \\ $$$$\mathrm{please}\:\mathrm{workings},\:\mathrm{how}\:\mathrm{is}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{15} \\ $$

Question Number 12590 Answers: 0 Comments: 0

In a basic version of pocker , each players is dealth 5 cards from a standard 52 cards (no pocker). How many diferent 5 cards pocker hand are there ???

$$\mathrm{In}\:\mathrm{a}\:\mathrm{basic}\:\mathrm{version}\:\mathrm{of}\:\mathrm{pocker}\:,\:\mathrm{each}\:\mathrm{players}\:\mathrm{is}\:\mathrm{dealth}\:\mathrm{5}\:\mathrm{cards}\:\mathrm{from}\:\mathrm{a}\:\mathrm{standard} \\ $$$$\mathrm{52}\:\mathrm{cards}\:\left(\mathrm{no}\:\mathrm{pocker}\right).\:\mathrm{How}\:\mathrm{many}\:\mathrm{diferent}\:\mathrm{5}\:\mathrm{cards}\:\mathrm{pocker}\:\mathrm{hand}\:\mathrm{are}\:\mathrm{there}\:??? \\ $$

Question Number 12218 Answers: 1 Comments: 3

5!! = ?

$$\mathrm{5}!!\:=\:? \\ $$

Question Number 12087 Answers: 1 Comments: 0

In how many ways can 10 objects be split into two groups containing 4 and 6 objects respectively ?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{10}\:\mathrm{objects}\:\mathrm{be}\:\mathrm{split}\:\mathrm{into}\:\mathrm{two}\:\:\mathrm{groups}\:\mathrm{containing}\: \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{6}\:\mathrm{objects}\:\mathrm{respectively}\:? \\ $$

Question Number 11855 Answers: 1 Comments: 0

Ten men are present at a club. In how many ways can four be chosen to play bridge if two men refuse to sit at the same table.

$$\mathrm{Ten}\:\mathrm{men}\:\mathrm{are}\:\mathrm{present}\:\mathrm{at}\:\mathrm{a}\:\mathrm{club}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{four}\:\mathrm{be}\:\mathrm{chosen}\:\mathrm{to} \\ $$$$\mathrm{play}\:\mathrm{bridge}\:\mathrm{if}\:\mathrm{two}\:\mathrm{men}\:\mathrm{refuse}\:\mathrm{to}\:\mathrm{sit}\:\mathrm{at}\:\mathrm{the}\:\mathrm{same}\:\mathrm{table}. \\ $$

Question Number 11425 Answers: 1 Comments: 0

Question Number 11384 Answers: 1 Comments: 0

Out of 5 accountants and 7 bankers, a committee consisting of 2 accountants and 3 bankers is to be formed. In how many ways can this be done if (a) Any acountant and any bankers must be included (b) One particular banker must be included (c) 2 accountant cannot be in the committee

$$\mathrm{Out}\:\mathrm{of}\:\mathrm{5}\:\mathrm{accountants}\:\mathrm{and}\:\mathrm{7}\:\mathrm{bankers},\:\mathrm{a}\:\mathrm{committee}\:\mathrm{consisting}\:\mathrm{of}\: \\ $$$$\mathrm{2}\:\mathrm{accountants}\:\mathrm{and}\:\mathrm{3}\:\mathrm{bankers}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{this} \\ $$$$\mathrm{be}\:\mathrm{done}\:\mathrm{if} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Any}\:\mathrm{acountant}\:\mathrm{and}\:\mathrm{any}\:\mathrm{bankers}\:\mathrm{must}\:\mathrm{be}\:\mathrm{included} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{One}\:\mathrm{particular}\:\mathrm{banker}\:\mathrm{must}\:\mathrm{be}\:\mathrm{included} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{2}\:\mathrm{accountant}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{in}\:\mathrm{the}\:\mathrm{committee} \\ $$

Question Number 11382 Answers: 0 Comments: 2

In how many ways can 24 different articles be divided into groups of 12, 8 and 4 articles respectively

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{24}\:\mathrm{different}\:\mathrm{articles}\:\mathrm{be}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{groups}\:\mathrm{of} \\ $$$$\mathrm{12},\:\mathrm{8}\:\mathrm{and}\:\mathrm{4}\:\mathrm{articles}\:\mathrm{respectively} \\ $$

Question Number 10029 Answers: 0 Comments: 1

The first 3 runners in 100m race were clocked 9.5s, 10s, 10.5s respectively. How far is the first from the third runners when the first runner reached the finished line.

$$\mathrm{The}\:\mathrm{first}\:\mathrm{3}\:\mathrm{runners}\:\mathrm{in}\:\mathrm{100m}\:\mathrm{race}\:\mathrm{were}\:\mathrm{clocked} \\ $$$$\mathrm{9}.\mathrm{5s},\:\mathrm{10s},\:\mathrm{10}.\mathrm{5s}\:\mathrm{respectively}.\:\mathrm{How}\:\mathrm{far}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{first}\:\mathrm{from}\:\mathrm{the}\:\mathrm{third}\:\mathrm{runners}\:\mathrm{when}\:\mathrm{the}\:\mathrm{first}\: \\ $$$$\mathrm{runner}\:\mathrm{reached}\:\mathrm{the}\:\mathrm{finished}\:\mathrm{line}. \\ $$

Question Number 9747 Answers: 1 Comments: 0

Find the coefficient of the term independent of x in the expansion of (((x + 1)/(x^(2/3) − x^(1/3) + 1)) − ((x − 1)/(x − x^(1/2) )))^(10)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{term}\:\mathrm{independent} \\ $$$$\mathrm{of}\:\mathrm{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\frac{\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}^{\mathrm{2}/\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{1}/\mathrm{3}} \:+\:\mathrm{1}}\:−\:\frac{\mathrm{x}\:−\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{x}^{\mathrm{1}/\mathrm{2}} }\right)^{\mathrm{10}} \\ $$

Question Number 9573 Answers: 0 Comments: 1

If n is positive integer prove that the cofficient of x^(2 ) and x^3 in the expansion of (x^2 +2x+2)^n are 2^(n−1) .n^2 and 2^(n−1) n(n−1)(1/3).

$${If}\:{n}\:{is}\:{positive}\:{integer}\:{prove}\:{that}\: \\ $$$${the}\:{cofficient}\:{of}\:{x}^{\mathrm{2}\:} {and}\:{x}^{\mathrm{3}} \:{in}\:{the}\: \\ $$$${expansion}\:{of}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)^{{n}} \:{are}\:\mathrm{2}^{{n}−\mathrm{1}} .{n}^{\mathrm{2}} \\ $$$${and}\:\mathrm{2}^{{n}−\mathrm{1}} {n}\left({n}−\mathrm{1}\right)\frac{\mathrm{1}}{\mathrm{3}}. \\ $$

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