CBSE Questions for Class 11 Engineering Maths Permutations And Combinations Quiz 8 - MCQExams.com

If the letters of the word "VARUN" are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is?
  • $$98$$
  • $$99$$
  • $$100$$
  • $$101$$
Garlands are formed using 6 red roses and 6 yellow roses of different sizes. The number of arrangements in garland which have red roses and yellow roses come alternately is
  • $$5!\times 6!$$
  • $$6! \times 6!$$
  • $$\dfrac{5!}{2!}\times 6!$$
  • $$2(6!\times 6!)$$
The value of  $$\sum\limits_{r = 0}^{10} {\left( {\matrix{   {10}  \cr    r  \cr } } \right)} \left( {\matrix{   {15}  \cr    {14 - r}  \cr } } \right)$$ is equal to
  • $${}^{25}{C_{12}}$$
  • $${}^{25}{C_{15}}$$
  • $${}^{25}{C_{10}}$$
  • $${}^{25}{C_{11}}$$
The product of five consecutive numbers is always divisible by ?
  • $$60$$
  • $$12$$
  • $$120$$
  • $$72$$
If $$f(x)=1-x+x^2-x^3+....-x^{15}+x^{16}-x^{17}$$, then the coefficient of $$x^2$$ in $$f(x-1)$$ is?
  • $$826$$
  • $$816$$
  • $$822$$
  • None of these
The value of $${ _{  }^{ 13 }{ C } }_{ 2 }+{ _{  }^{ 13 }{ C } }_{ 3 }+{ _{  }^{ 13 }{ C } }_{ 4 }+....+{ _{  }^{ 13 }{ C } }_{ 13 }$$ is 
  • $${2}^{13}-13$$
  • $${2}^{13}-14$$
  • an odd number $$\ne {2}^{13}-12$$
  • an even number $$\ne {2}^{13}-14$$
A person tries to form as many different parties as he can, out of his $$20$$ friends. Each party should consist of the same number. How many friends should be invited at a time? In how many of these parties would the same friends be found?
  • 92378
  • 92364
  • 92376
  • 92391
A question paper on mathematics consists of twelve questions divided into three parts A, B and C, each containing four questions. In how many ways can an examinee answer five questions, selecting atleast one from each part.
  • $$624$$
  • $$208$$
  • $$2304$$
  • None
$$\frac{{({C_0} + {C_1})({C_1} + {C_2})({C_2} + {C_3}).....({C_{n - 1}} + {C_n})}}{{{C_0} + {C_1}({C_2}......{C_n})}} = $$
  • $$\frac{{{{(n + 1)}^n}}}{{n!}}$$
  • $$\frac{{{{(n + 2)}^n}}}{{n!}}$$
  • $$\frac{{{{(n + 1)}^{n - 1}}}}{{n!}}$$
  • $$\frac{{{{(n - 1)}^n}}}{{n!}}$$
In an examination there are three multiple choice questions and each question has $$4$$ choices. Number of ways in which a student can fail to get all answer correct is?
  • $$11$$
  • $$12$$
  • $$27$$
  • $$63$$
$$7$$ boys and $$8$$ girls have to sit in a row on $$15$$ chairs numbered from $$1$$ to $$15$$ then?
  • Number of ways boys and girls sit alternately is $$8!7!$$
  • Number of ways boys and girls sit alternately is $$2(8!7!)$$
  • The number of ways in which first and fifteenth chair are occupied by boys and between any two boys an even number of girls sit is $$^9C_4$$ $$8!7!$$
  • The number of ways in which first and last seat are occupied by boys and between any two boys an even number of girls sit is $$(2{^9C_4}8!7!)$$.
Number of different paths of shortest distance from A to B in the grid which do not pass through M.
1148217_b3e1a3d279c54885836a6521698d4225.png
  • $$462$$
  • $$262$$
  • $$442$$
  • $$^{11}C_5-[^6C_3\times^5C_2]$$
State following are True or False

If m=n=p and the groups have identical qualitative characterstic then the number of groups $$=\dfrac { (3n)! }{ n!n!n!3! } $$
Note : If 3n different things are to be distributed equally three people then the number of ways$$=\dfrac { (3n)! }{ { (n!) }^{ 3 } } $$
  • True
  • False
A is a set containing n elements. A  subset P of A is chosen. The set A is reconstructed by replacing the element of P.A subset Q of A is again chosen. The number of way of choosing P and Q so that P Q =$$\phi $$ is :- 
  • $$2^{2n}-^{2n}C_{n}$$
  • $$2^{n}$$
  • $$2^{n}-1$$
  • $$3^{n}$$
A box contains 5 pairs of shoes. If 4 shoes are selected, then the number of ways in which exactly one pair of shoes obtained is :
  • 120
  • 140
  • 160
  • 180
Let $$5 < n_1 < n_2 < n_3 < n_4$$ be integers such that $$n_1+n_2+n_3+n_4=35$$. The number of such distinct arrangements $$(n_1, n_2, n_3, n_4)$$.
  • $$^{38}C_3$$
  • $$^8C_3$$
  • $$5$$
  • $$6$$
Let $$(1+x)^n=C_0+C_1x+C_2x^2+....+C_nX^n$$.(where $$C_r=$$ $$^{n}C_r$$). On the basis of information, answer the following question.
$$2(C_2)-4(C_4)+6(C_6)$$_______ is?
  • $$(\sqrt{2})^{\dfrac{(n-1)}{2}}$$
  • $$2^{\dfrac{(n-1)}{2}}.n.\sin\left(\dfrac{(n-1)\pi}{4}\right)$$
  • $$\sin \dfrac{n\pi}{4}$$
  • $$2^{\dfrac{(n-1)}{2}}.n.\sin\dfrac{n\pi}{4}$$
Given $$4$$ flags of different colours, how many different signals can be generated. If a signal requires the use of $$2$$ flags one below the other?
  • $$4$$
  • $$3$$
  • $$12$$
  • $$1$$
There are $$k$$ different books and $$l$$ copies of each in a college library. The number of ways in which a student can make a selection of one or more books is
  • $$(k+1)^{l}$$
  • $$(l+1)^{k}$$
  • $$(k+1)^{l}-1$$
  • $$(l+1)^{k}-1$$
There are $$4$$ letter boxes in a post office. In how many ways can a man post $$8$$ distinct letters? 
  • $$4\times 8$$
  • $${8}^{4}$$
  • $${4}^{8}$$
  • $$P\left (8,4\right)$$
$$4$$ buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, then the total possible ways are 
  • $$12$$
  • $$16$$
  • $$4$$
  • $$8$$
Number of five-digit numbers divisible by 5 that can be formed from the digits $$0,1, 2, 3, 4, 5$$ without repetition of digits are
  • $$240$$
  • $$360$$
  • $$148$$
  • $$216$$
Ten persons, amongst whom are $$A$$,$$B$$ and $$C$$ to speak at a function. The number of ways in which it can be done if $$A$$ wants to speak before $$B$$ AND $$B$$ wants to speak before $$C$$ is 
  • $$\dfrac{{10!}}{6}$$
  • $$3!$$ $$7!$$
  • $$^{10}{P_3}.7!$$
  • none of these
There are n locks and n matching keys. If all the locks and keys are to be perfectly matched, find the maximum number of trails required to open a lock.
  • $$^{n}C_{2}$$
  • $$\sum _{ k=2 }^{ n }{ \left( k+2 \right) } $$
  • $$\dfrac{n(n+1)}{2}$$
  • $$^{n+1}C_{2}$$
The number of ways in which a mixed doubles tennis game can be arranged between 10 players consisting of 6 men and 4 women is . 
  • 180
  • 90
  • 48
  • 12
Let $$p=\dfrac{1}{1\times2}+\dfrac{1}{3\times4}+\dfrac{5}{1\times6}+.......+\dfrac{1}{2013\times2014}$$ and $$Q=\dfrac{1}{1008\times2014}+\dfrac{1}{1009\times2013}+.........+\dfrac{1}{2014\times1008}$$
then $$\dfrac{P}{Q}=$$
  • $$2013$$
  • $$2014$$
  • $$1511$$
  • $$2$$
The number of intersection points of diagonals of $$2009$$ sides polygon, which lie  inside the polygon.
  • $$^{2009}C_4$$
  • $$^{2009}C_2$$
  • $$^{2008}C_4$$
  • $$^{2008}C_2$$
Let the eleven letters, $$A, B, ....K$$ denote an artbitrary permutation of the integers $$(1,2,....11)$$, then $$(A-1)(B-2)(C-3)...(K-11)$$ is
  • Necessarily zero
  • Always odd
  • Always evem
  • None of these

$$1+2+3+......t_n= ^{n+1}P_2$$
  • True
  • False
Consider all permutations of the letters of the word MORADABAD.
The number of permutations which contain the word BAD is:
  • $$21 \times 5!$$
  • $$7 \times 5!$$
  • $$6 \times 5!$$
  • $$2 \times 5!$$
The number of permutation of the letters of the word $$HINDUSTAN$$ such that neither the pattern $$'HIN'$$ nor $$'DUS'$$ nor $$'TAN'$$ appears, are :
  • $$166674$$
  • $$169194$$
  • $$166680$$
  • $$181434$$
In how many ways atleast one horse and atleast one dog can be selected out of eight horses and seven dogs.
  • $${2}^{15}-2$$
  • $${2}^{15}-1$$
  • $$({2}^{8}-1)({2}^{7}-1)$$
  • $${ _{ }^{ 15 }{ C } }_{ 2 }$$
Determine $$n$$, if $$^ { 2 n } \mathrm { C } _ { 3 } : ^ { n } \mathrm { C } _ { 3 } = 12 : 1$$
  • $$2$$
  • $$3$$
  • $$4$$
  • $$5$$
The total number of ways of arranging the letters $$AAABBBCCDEF$$ in a row such that letters $$C$$ are separated from one another is
  • $$277200$$
  • $$138600$$
  • $$453600$$
  • none of these
$$^{404}C_{4}-^{4}C_{1}^{303}C_{4}+^{4}C_{2}^{202}C_{4}-^{404}C_{4}^{101}C_{4}$$ is equal to
  • $$(401)^{4}$$
  • $$(101)^{4}$$
  • $$0$$
  • $$(201)^{4}$$
The number of ways in which the letters of the word $$"ARRANGE"$$ can be permuted such that $$R's$$ occur together is 
  • $$\dfrac{!7}{!2!2}$$
  • $$6!$$
  • $$\dfrac{6!}{2!}$$
  • none of these
The number of positive integral solutions of the equation $$x _ { 1 } x _ { 2 } x _ { 3 } x _ { 4 } x _ { 5 } = 1050$$ is
  • 1800
  • 1600
  • 1400
  • 1875
If $$(1+x+x^2)^n=\displaystyle\sum^{2n}_{r=0}a_rx^r$$, then $$a_0a_{2r}-a_1a_{2r+1}+a_2a_{2r+2}-....=?$$
  • $$a_r$$
  • $$a_{n-2r}$$
  • $$a_{n+r}$$
  • $$a_{2r}$$
If repetitions are not allowed, the number of numbers consisting of $$4$$ digits and divisible by $$5$$ and formed out of $$0,1,2,3,4,5,6$$ is 
  • $$220$$
  • $$240$$
  • $$370$$
  • $$588$$
Number of cyphers at the end of 202$$\mathrm { C } _ { 1001 }$$ is ________.
  • 0
  • 1
  • 2
  • None
Letters of the word $$MATHEMATICS$$ are arranged in all the possible ways, in how many words letter $$C$$ is between $$S$$ and $$H$$(these three letter are not necessary together)? 
  • $$(11!)/(2!)(2!)(2!)$$
  • $$(11!)/(3!)(2!)(2!)$$
  • $$(11!)/(3!)(3!)(2!)$$`
  • $$None\ of\ these$$
If $${ S }_{ n }={ C }_{ 0 }{ C }_{ 1 }+{ C }_{ 1 }{ C }_{ 2 }+.......+{ C }_{ n-1 }{ C }_{ n }$$ and $$\frac { { S }_{ n+1 } }{ { S }_{ n } } =\frac { 15 }{ 4 } $$, then value of n
  • 3, 7
  • 2, 4
  • 1, 3
  • 1, 2
If $$x+y=1$$, then $$\displaystyle\sum^n_{r=0}r\cdot {^{n}C_r}x^r\cdot y^{n-r}=?$$
  • $$1$$
  • n
  • nx
  • ny
If $$\displaystyle\sum^{n-r}_{k=1}$$ $$^{n-k}C_r={^{x}C_y}$$ then?
  • $$x=n+1; y=r$$
  • $$x=n; y=r+1$$
  • $$x=n; y=r$$
  • $$x=n+1; y=r+1$$
The number of permutations of letters of the word "PARALLAL" atken four at a time must be, 
  • $$216$$
  • $$244$$
  • $$286$$
  • $$1680$$
The value of $$\sum _{ r=0 }^{ s }{ \sum _{ s=1 }^{ n }{  }  } ^nC_s.$$ $$^sC_r$$ (where $$r\le s$$ is )
  • $$3^n$$
  • $$n.3^{n-1}$$
  • $$n.3^{n-1}-1$$
  • $$3^n-1$$
If $$\displaystyle \frac{1}{{^4{C_n}}} = \frac{1}{{^5{C_n}}} + \frac{1}{{^6{C_n}}}$$, then $$n=$$
  • $$3$$
  • $$2$$
  • $$1$$
  • $$0$$
There are n identical red balls & m identical green balls. The number of different linear arrangements consisting of "n red balls but not necessarily all the green balls" is $${^{x}C_y}$$ then?
  • $$x=m+n, y=m$$
  • $$x=m+n+1, y=m$$
  • $$x=m+n+1, y=m+1$$
  • $$x=m+n. y=n$$
The exponent of 11 in $$^{200}C_{125}$$ is 
  • 2
  • 1
  • 4
  • 5
The number of ways of 3 scholarship of unequal value be awarded to 17 candidates, Such that no candidate gets more than one scholarship is 
  • $$^{ 17 }{ C }_{ 3 }$$
  • $${ 17 }^{ 3 }$$
  • $${ 3 }^{ 17 }$$
  • $$^{ 17 }P_{ 3 }$$
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