CBSE Questions for Class 11 Engineering Maths Permutations And Combinations Quiz 12 - MCQExams.com

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?
  • $$25$$
  • $$41$$
  • $$22$$
  • $$42$$
The number of ways in which four letters can be selected from the word 'DEGREE' is
  • $$7$$
  • $$6$$
  • $$\dfrac{6!}{3!}$$
  • None of these
If $$^{15}C_{3r} = ^{15}C_{r+3}$$, then find the value of $$r$$:
  • $$3$$
  • $$2$$
  • $$0$$
  • $$1$$
The value of $$\sum^{10}_{r=0}\begin{pmatrix}10\\r\end{pmatrix}\begin{pmatrix}15\\14-r\end{pmatrix}$$ is equal to
  • $$^{25}C_{12}$$
  • $$^{25}C_{15}$$
  • $$^{25}C_{10}$$
  • $$^{25}C_{14}$$
Two lines intersect at $$O$$. Points $$A_{i}$$ and $$B_{i} (i = 1, 2, ...., n)$$ are taken on these two lines respectively, the number of triangles that can be drawn with the help of these $$2n + 1$$ points is
  • $$n$$
  • $$n^{2}$$
  • $$n^{3}$$
  • None of these
Value of $$\sum _{ k=0 }^{ n }{ { _{  }^{ k }{ C } }_{ n }\sin { \left( kx \right)  } \cos { \left( n-k \right)  }  } $$ is
  • $${ 2 }^{ n-1 }\left( \sin { nx } \right) $$
  • $${ 2 }^{ n }\sin { \left( nx \right) } $$
  • $${ 2 }^{ n }\cos { \left( nx \right) } $$
  • none of these
In a triangle $$ABC$$, the value of the expression $$\displaystyle \sum_{r = 0}^{n}\ ^{n}C_{r}a^{r}.b^{n - r}.\cos (rB - (n - r)A)$$ is equal to
  • $$c^{n}$$
  • zero
  • $$a^{n}$$
  • $$b^{n}$$
Seven person $$P_1,P_2......, P_7$$ initially seated at chairs $$C_1,C_2,.....C_7$$ respectively.They all left there chairs simultaneously for hand wash. Now in how many ways they can again take seats such that no one sits on his own seat and $$P_1$$, sits on $$C_2$$ and $$P_2$$ sits on $$C_3$$ ?
  • $$52$$
  • $$53$$
  • $$54$$
  • $$55$$
If $$m$$ denotes the number of $$5$$ digit numbers if each successive digits are in their descending order of magnitude and $$n$$ is the corresponding figure. When the digits and in their ascending order of magnitude then $$(m-n)$$ has the value
  • $$ { _{ }^{ 9 }{ C } }_{ 4 }$$
  • $${ _{ }^{ 9 }{ C } }_{ 5 }$$
  • $${ _{ }^{ 10 }{ C } }_{ 3 }$$
  • $$ { _{ }^{ 9 }{ C } }_{ 3 }$$
There are $$2$$ identical white balls, $$3$$ identical red balls and $$4$$ green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is
  • $$6(7! - 4!)$$
  • $$7(6! - 4!)$$
  • $$8! - 5!$$
  • None
Two classrooms A and B having capacity of $$25$$ and $$(n-25)$$ seats respectively. $$A_n$$ denotes the number of possible seating arrangements of room $$'A'$$, when 'n' students are to be seated in these rooms, starting from room $$'A'$$ which is to be filled up to its capacity. If $$A_n-A_{n-1}=25!(^{49}C_{25})$$ then 'n' equals:
  • $$50$$
  • $$48$$
  • $$49$$
  • $$51$$
If $$\displaystyle \sum_{k = 1}^{n = 1} \ ^{n - k}C_{r} = ^{x}C_{y}$$ then
  • $$x = n + 1; y = r$$
  • $$x = n; y = r + 1$$
  • $$x = n; y = r$$
  • $$x = n + 1; y = r + 1$$
If $${ \left( 1+x \right)  }^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r } } $$ and $${ b }_{ r }=1+\cfrac { { a }_{ r } }{ { a }_{ r-1 } } $$ and $$\prod _{ r=1 }^{ n }{ { b }_{ r }=\cfrac { { \left( 101 \right)  }^{ 100 } }{ 100! }  } $$, then $$n$$ equals to:
  • $$99$$
  • $$100$$
  • $$101$$
  • None of these
Sum of series : $$\sum_\limits{r=1}^n (r^2+1)( r! )$$ is _______
  • (n+1) !
  • (n+2) ! - 1
  • n(n + 1) !
  • None of these
There are infinite, alike, blue, red, green and yellow balls. Find the number of ways to select $$10$$ balls.
  • $$286$$
  • $$84$$
  • $$715$$
  • None of these.
Let $$(1+x)^n=C_0+C_1x+C_2x^2+.....+C_nX^n$$. (where $$C_r=^nC_r$$)
On the basis of given information, answer the following question.
$$C_1-3(C_3)+5(C_5)-7(C_7)+$$_______ is?
  • $$n.2^{\left(\dfrac{n-1}{2}\right)}$$
  • $$n\left(2^{\dfrac{(n-1)}{2}}\right)\cos \dfrac{n\pi}{4}$$
  • $$n(2^{(n-1)})$$
  • $$n.2^{\dfrac{(n-1)}{2}}\cos\left(\dfrac{(n-1)\pi}{4}\right)$$
Maximum number of common chords of a parabola and a circle can be equal to
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
In a certain examination paper, there are $$n$$ question. For $$j = 1, 2....n$$, there are $$2^{n-j}$$ students who answered $$j$$ or more questions wrongly. If the total number of wrong answers is $$4095$$, then the value of $$n$$ is:
  • $$12$$
  • $$11$$
  • $$10$$
  • $$9$$
How many $$10-digit$$ numbers can be formed by using the digits $$1$$ and $$2$$?
  • $$^{10}{P}_{2}$$
  • $$^{10}{C}_{2}$$
  • $${2}^{10}$$
  • $$10\ !$$
The value of $$\displaystyle\sum^{10}_{r=0}(r)$$ $$^{20}C_r$$ is equal to?
  • $$20(2^{18}+{^{19}}C_{10})$$       
  • $$10(2^{18}+{^{19}}C_{10})$$
  • $$20(2^{18}+{^{19}}C_{11})$$
  • $$10(2^{18}+{^{19}}C_{11})$$
What is the value of $$^nC_n$$?
  • zero
  • $$1$$
  • $$n$$
  • $$n!$$
Let $$S$$ be the set of $$6$$ digits numbers $$a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$$ (all digits distinct) where $$a_{1}>a_{2}>a_{3}<a_{4}>a_{5}<a_{6}$$. Then $$n(S)$$ is equal to
  • $$210$$
  • $$2100$$
  • $$4200$$
  • $$420$$
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is 
  • 140
  • 196
  • 280
  • 346
A library has $$'a'$$ copies of one book, $$'b'$$ copies of each of two books, $$'c'$$ copies of each of three books, and single copy each of $$'d'$$. The total number of ways in which these books can be arranged in a row is
  • $$\dfrac{(a+b+c+d)!}{a!b!c!}$$
  • $$\dfrac{(a+2b+3c+d)!}{a!(b!)^{2}(c!)^{3}}$$
  • $$\dfrac{(a+2b+3c+d)!}{a!b!c!}$$
  • none of these
If $$S_n=\displaystyle\sum^{n}_{r=0}\dfrac{1}{^{n}C_r}$$ and $$t_n=\displaystyle\sum^n_{r=0}\dfrac{r}{^{n}C_r}$$, then $$\dfrac{t_n}{S_n}$$ is equal to?
  • $$\dfrac{1}{2}n$$
  • $$\dfrac{1}{2}n-1$$
  • $$n-1$$
  • $$\dfrac{2n-1}{2}$$
$$2C_0 + \dfrac{2^2}{2} C_1 + \dfrac{2^3}{3} C_2 +........+ \dfrac{2^{11}}{11} C_{10}$$ is equal to
  • $$\dfrac{2^{11} -1}{11}$$
  • $$\dfrac{3^{11} -1}{11}$$
  • $$\dfrac{3^{11} -1}{12}$$
  • $$\dfrac{3^{11} +1}{11}$$
The number of 5 letter words formed  using letters of word "CALCULUS" is  
  • $$280$$
  • $$15$$
  • $$1110$$
  • $$56$$
If $$^{2n}C_{3}:^{n}C_{2}::44:1$$, then the value of $$n$$ is
  • $$17$$
  • $$6$$
  • $$11$$
  • $$none\ of\ these$$
If m denotes the number of $$5$$ digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure, when the digits are in their ascending order of magnitude then $$(m-n)$$ has the value?
  • $$^{10}C_4$$
  • $$^9C_5$$
  • $$^{10}C_3$$
  • $$^9C_3$$
$$10$$ IIT and $$2$$ NIT students sit at random in a row, and then number of ways in which exactly $$3$$ IIT students sit between $$2$$ NIT students is
  • $$16\times 10!$$
  • $$15\times 10!$$
  • $$10\times 16!$$
  • $$None\ of\ these$$
$${ if}^{ n }{ c }_{ 10 }{ = }^{ n }{ c }_{ 14 }$$ then the value of n  is equal to
  • 14
  • 24
  • 34
  • 44
$${ if }^{ n }{ c }_{ r }{ + }^{ n }{ c }_{ r+1 }={ n+1 }_{ { C }_{ x } }$$ then the value of x is equal to
  • r
  • r + 1
  • r - 1
  • 2r
The sum $$^{ n }{ C }_{ 0 }+^{ n }{ C }_{ 1 }+^{ n }{ C }_{ 2 }+.......+^{ n }{ C }_{ n }$$ is equal to 
  • $$\frac { 2.4.6.........2n }{ n\quad ! } $$
  • $${ n }^{ n }$$
  • $$n!$$
  • $${ 3 }^{ n }$$
If $$\displaystyle\sum^m_{k=1}(k^2+1)k!=1999(2000!)$$, then m is?
  • $$1999$$
  • $$2000$$
  • $$2001$$
  • $$2002$$
If $$n\ \in\ N$$ & $$n$$ is even, then $$\dfrac {1}{1\ .\ (n-1)\ !}+\dfrac {1}{3\ !(n-3)\ !}+\dfrac {1}{5\ !\  (n-5)\ !}+....+\dfrac {1}{(n-1)\ !\ 1\ !}=$$ 
  • $$2^{n}$$
  • $$\dfrac {2^{n-1}}{n\ !}$$
  • $$2^{n}n\ !$$
  • $$none\ of\ these$$
The value of $$\sum _{ r=0 }^{ n }{ \sum _{ s=0 }^{ n }{ (r+s){ C }_{ r }{ C }_{ s } }  } ,$$ is 
  • $$n.2^{2n}$$
  • $$n(n-1)2^{n-2}$$
  • $$n(n+1)2^n$$
  • $$2^{2n}$$
if$$^{ 2017 }{ c }_{ 0 }{ + }^{ 2017 }{ c }_{ 1 }{ + }^{ 2017 }{ c }_{ 2 }+.....{ + }^{ 2017 }{ c }_{ 1008 }={ \lambda  }^{ 2 }(\lambda >0),$$ then remainder when $$\lambda$$ is divided by 33 is
  • 8
  • 13
  • 17
  • 25
$${ if } ^ { n }{ C }_{ 3 }{ + }^{ n }{ C }_{ 4 }>{ n+1 }_{ C_{ 3 } }$$ then
  • n > 6
  • n > 7
  • n < 6
  • none of these
Value of $$\sum _{ r=1 }^{ n }{ \left( \sum _{ m=0 }^{ r }{ {  }_{  } }  \right)  } ^nC_r,^rC_m)$$ is equal to 
  • $$2^n-1$$
  • $$3^n-1$$
  • $$3^n-2^n$$
  • $$None$$ $$of$$ $$these$$
If $$(1+x)^n=C_0+C_1x+C_2x^2+....+C_nX^n$$, then the value of $$C^2_0+\dfrac{C^2_1}{2}+\dfrac{C^2_2}{3}+....+\dfrac{C^2_n}{n+1}$$ is equal to?
  • $$\dfrac{(2n-1)!}{\{(n+1)!\}^2}$$
  • $$\dfrac{(2n-1)!}{(n+1)!}$$
  • $$\dfrac{2n+1)!}{\{(n+1)!\}^2}$$
  • $$\dfrac{(2n)!}{(n+1)!}$$
If $$(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+...+C_{n}x^{n}$$ then the value of $$1^{2}C_{1}+2^{2}C_{2}+3^{2}C_{3}+...+n^{2}C_{n}$$ is
  • $$n(n+1)2^{n-2}$$
  • $$n(n+1)2^{n-1}$$
  • $$n(n+1)2^{n}$$
  • None of these
If n is odd natural number, then $$\sum _{ r=0 }^{ n }{ \cfrac { { (-1) }^{ r } }{ ^{ n }{ C }_{ r } }  } $$ equals
  • 0
  • 1/n
  • $$n/{ 2 }^{ n }$$
  • None of these
There are $$n$$ points on a circle. The number of staight lines formed by joining them is equal to
  • $$^{n}C_{2}$$
  • $$^{n}P_{2}$$
  • $$^{n}C_{2}-1$$
  • $$none\ of\ these$$
The number of ways $$5$$ identical balls can be distributed into $$3$$ different boxes so that no box remains empty.
  • $$^{4}C_2$$
  • $$6$$
  • $$^{4}C_1$$
  • $$^{4}C_3$$
The number of words which can be made out of the letters of the word $$'MOBILE'$$ when consonants always occupy odd places is _______.
  • $$20$$
  • $$36$$
  • $$30$$
  • $$720$$
In a set of lottery Tickets $$7$$ carry prizes and $$25$$ are blank. If three tickets are drawn then the probability to get a prize is?
  • $$\dfrac{{^{7}C_3}}{{^{32}C_3}}$$
  • $$\dfrac{{^{25}C_3}}{{^{32}C_3}}$$
  • $$1-\dfrac{{^{25}C_3}}{{^{32}C_3}}$$
  • Cannot be decided
The sum of the numbers formed by taking all the digit at a from 0, 2, 3, 4 is 
  • 57996
  • 75669
  • 99657
  • 57699
Number of six digit numbers whose sum of the digits is $$49$$ are 
  • $$^{12}C_{5}$$
  • $$6^{10}$$
  • $$^{10}C_{5}$$
  • $$1200$$
The letters of the word 'VICTORY' are arranged in all possible ways, and the words thus obtained are arranged as in a dictionary. Then the rank of given word is 
  • 3733
  • 5309
  • 5040
  • 3732
$$20$$ soldiers are standing in a row and their captain want to send $$7$$ out of them for a mission. In how many ways can captain select them such that at least one soldier find the soldier  next to him is also selected.   
  • $$^{20}C_{7}$$
  • $$^{14}C_{7}$$
  • $$^{20}C_{7}-^{13}C_{7}$$
  • $$^{20}C_{7}-^{14}C_{7}$$
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