CBSE Questions for Class 11 Engineering Maths Permutations And Combinations Quiz 3 - MCQExams.com

If $$ ^nC_3 : ^{2n-1}C_2  =$$ $$8:15$$, then $$n$$ $$=$$
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
The least value of the natural number '$$n$$' satisfying $$c(n,5) + c(n,6) > c(n+1,5)$$
  • 10
  • 12
  • 13
  • 11
 Match the following :
 $$ A)^{ n }C_{ 0 }+^{ n }C_{ 1 }+^{ n }C_{ 2 }+....+^{ n }C_{ n }$$    1)$$^{ (n+1) }P_{ r } $$
 $$ B) { \dfrac { P_{ r } }{ P_{ (r-1) } }  } \;\;\; \; $$  2) $$2^n $$

 $$ C) ^nP_r + r   ^n .P_{(r-1)}$$ 3)$$ ^{(n+1)}P_{r} $$
 $$ D) ^nP_n$$4)$${n-r+1}$$
  5)$$ n! $$

                     
  • $$A-2, B-4,C-1,D-5$$
  • $$A-1, B-4,C-2,D-5$$
  • $$A-2, B-4,C-5,D-1$$
  • $$A-1, B-3,C-5,D-2$$
If $$ ^nC_r + ^nC_{r+1} = ^{(n+1)}C_x$$, then  $$x= $$
  • $$r$$
  • $$r-1$$
  • $$n$$
  • $$r+1$$
If $$^nP_r = 840, ^nC_r = 35$$, then $$n$$ is equal to:
  • 6
  • 7
  • 8
  • 9
If $$C (2n, 3) : C (n, 2)$$ $$=$$ $$12 : 1$$, then $$n$$ $$ = $$
  • $$4$$
  • $$5$$
  • $$6$$
  • $$8$$
If $$ ^{n+2}C_8 : ^{n-1}P_4  =  133:48$$, then $$n$$ $$=$$
  • $$17$$
  • $$18$$
  • $$19$$
  • $$20$$
$$ \left(\:^{22}C_{5}+\:^{22}C_{4} \right)+\:^{23}C_{4}+\:^{24}C_{4}+\:^{25}C_{4} =  ?$$
  • $$ ^{22}C_5 $$
  • $$ ^{27}C_4 $$
  • $$ ^{26}C_4 $$
  • $$ ^{26}C_5 $$
Observe the following Lists

 List  I 

 List II 

$$ A)^{ { n } }C_{ r }+^{ n }C_{ r-1 }= $$

$$ 1)^{ n+1 }P_{ r } $$

$$ B) ^{ \dfrac {^{n} P_{ r } }{^{n} P_{ r-1 } } = } $$

$$ 2)\displaystyle \frac { n-r+1 }{ r } $$

$$ C) ^{n}P_r+r\ {}^{n}P_{r -1 }=$$

$$ 3) { n-r+1 } $$

$$ D) { \displaystyle \frac {^{n} C_r }{^{n} C_{ r-1 } } = } $$

$$ 4)n+r-1 $$


$$ 5)^{(n+1)}C_{ r } $$
  • A-5, B-3, C-2, D-1
  • A-5, B-3, C-1, D-2
  • A-2, B-4, C-3, D-1
  • A-5, B-4, C-2, D-1
If $$n={ ^{ m }{ C } }_{ 2 }$$, then the value of $${ ^{ n }{ C } }_{ 2 }$$ is given by
  • $${ ^{ m+1 }{ C } }_{ 4 }$$
  • $${ ^{ m-1 }{ C } }_{ 4 }$$
  • $${ ^{ m+2 }{ C } }_{ 4 }$$
  • none of these
If $$ 3.^{ (x+1) }C_{ 2 }+^{ 2 }P_{ 2 }.x=4.^{x}P_{ 2 },x\in N$$, then $$x= $$
  • $$2$$
  • $$4$$
  • $$5$$
  • $$3$$
Number of points having position vector $$a\hat{i}+b\hat{j}+c\hat{k},$$ where $$a,b,c\in \left\{1,2,3,4,5\right\}$$ such that $$2^a+3^b+5^c$$ is divisible by $$4$$ is
  • $$140$$
  • $$70$$
  • $$100$$
  • none of these 
Nine hundred distinct $$n$$ digit numbers are to be formed using only the three digits $$2,5$$ and $$7$$. the smallest value of $$n$$ for which this is possible is
  • $$4$$
  • $$8$$
  • $$7$$
  • $$9$$
The sum $$\sum_{i=0}^{m}\binom{10}{i}\binom{20}{m-i}$$ where $$\binom{p}{q} = 0$$; If (p<q) is maximum when m is
  • 5
  • 10
  • 15
  • 20
Arrange the following values of n in ascending order.
$$ A: ^nP_5= ^nP_6 \Rightarrow n = $$
$$ B: ^nP_{12}= ^nP_8 \Rightarrow n = $$
$$ C: ^nC_{(n-3)}= 10 \Rightarrow n = $$
$$ D: ^{(n+1)}P_5: ^nP_6  =1:2 \Rightarrow n = $$
  • CABD
  • CADB
  • ACBD
  • DBAC
The number of values of $$r$$ satisfying the equation $$ ^{39}C_{3r-1} - ^{39}C_{r^2} = ^{39}C_{r^2-1} - ^{39}C_{3r} $$ is:
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
There are $$n$$ different books and $$p$$ copies of each in a library. The number of ways in which one or more than one book can be selected is:
  • $$ p^n + 1 $$
  • $$ {(p+1)}^n - 1 $$
  • $$ {(p+1)}^n - p $$
  • $$ p^n $$
The number of quadratic expressions with the coefficients drawn from the set $$( 0, 1, 2, 3 )$$ is:
  • $$27$$
  • $$36$$
  • $$48$$
  • $$64$$
$$ ^nC_0+^{n+1}C_1+ ^{n+2}C_2+ ....+ ^{n+r}C_r = $$
  • $$ ^{n+r+1}C_r $$
  • $$ ^{n+r-1}C_r $$
  • $$ n $$
  • $$ ^{n+r+2}C_r $$
$$ ^nC_{r+1} + 2 ^nC_r + ^nC_{r-1}  =$$
  • $$ ^{n+1}C_{r} $$
  • $$ ^{n+2}C_{r} $$
  • $$ ^{n+2}C_{r+1} $$
  • $$ ^{n+2}C_{r+2} $$
If $$n$$ is an integer between $$0$$ and $$21$$, then the minimum value of $$n! (21-n)$$! is
  • $$9! 2!$$
  • $$10! 11!$$
  • $$20!$$
  • $$21!$$
If $$n$$ and $$r$$ are integers such that $$ 1 \le r \le n $$, then    $$n.^{n-1}C_{r - 1} = $$
  • $$^nC_r$$
  • $$n^nC_r$$
  • $$r^nC_r$$
  • $$(n-1)^nC_r$$
If $$r > 1$$, then $$\displaystyle  \frac{^nP_r}{^nC_r}  $$ is
  • is an integer
  • may be fraction
  • is an odd number
  • an even number
$$5$$ balls of different colours are to be kept in $$3$$ boxes of different sizes. Each box can hold all five balls. Number of ways in which the balls can be kept in the boxes so that no box remain empty is
  • $$60$$
  • $$90$$
  • $$150$$
  • $$200$$
The number of solutions of $$ ^{61}C_{n + 1} = ^{61}C_{2n - 1} $$ is
  • 3
  • 1
  • 2
  • 4
$$ ^{(k-1)}C_{(k-1)}+^{k}C_{(k-1)}+^{(k+1)}C_{(k-1)}+ .......+ ^{(n+k-2)}C_{(k-1)} $$ $$=?$$
  • $$ ^{(n+k)}C_k $$
  • $$ ^{(n+k+1)}C_k $$
  • $$ ^{(n+k)}C_{k-1} $$
  • $$ ^{(n+k-1)}C_k $$
$$ ^nP_r$$  and   $$^nC_r $$ are equal when:
  • $$n = r$$
  • $$n = r + 1$$
  • $$r = 1$$
  • $$n = r - 1$$
If $$ 10(^nC_2) = 3(^{n+1}C_3) $$, then $$n$$ $$=$$
  • 8
  • 9
  • 10
  • 11
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is incorrect and Reason is correct

 $$\displaystyle \frac{n_{C_{r}}}{n_{C_{r-1}}}=$$
  • $$\displaystyle \frac{n+r+1}{r}$$
  • $$\displaystyle \frac{n-r+1}{r}$$
  • $$\displaystyle \frac{n-r-1}{r}$$
  • $$\displaystyle \frac{n+r-1}{r}$$
In the next world cup of cricket, there will be $$12$$ teams, divided equally into two groups. Teams of each group will play a match against each other. From each group $$3$$ top teams will qualify for the next round. In this round, each team will play against other once. Four top teams from this round, where each team will play against the other three. Two top teams of this round will goto the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be
  • $$54$$
  • $$53$$
  • $$38$$
  • $$55$$
If $$ ^nC_{r} = 10, ^nC_{r+1}=45 $$ then, $$r$$ equals
  • 1
  • 2
  • 3
  • 4
The least value of $$n$$ so that  $$ ^nC_6 + ^nC_7 >  ^{n+1} C_6 $$ is
  • 13
  • 12
  • 11
  • 10
The least positive integral value  of $$x$$ which satisfies the in equality $$ ^{10}C_{x-1} >^{10}C_x $$ is
  • $$7$$
  • $$8$$
  • $$9$$
  • $$10$$
The number of positive integers satisfying the inequality $$ ^{(n+1)}C_{n-2}- ^{n+1}C_{n-1}\le$$ $$100$$ is
  • 9
  • 8
  • 5
  • 7
Let x.y.z$$=$$105 where $$ x, y, z \in N $$. Then number of ordered triplets (x, y, z) satisfying the given equation is
  • 15
  • 27
  • 6
  • 33
If $$ P_n $$  denotes the product of all the coefficients in the expansion of $${(1+x)}^n$$ and $$ 9! P_{(n+1)} =10\ ^{9}P_n $$. Then $$n =$$
  • $$10$$
  • $$9$$
  • $$19$$
  • $$11$$
Let $$\mathrm{S}=\{1,2,3,4\}$$. The total number of unordered pairs of disjoint subsets of $$\mathrm{S}$$ is equal to
  • 25
  • 34
  • 42
  • 41
There are $$4$$ candidates for a Natural science scholarship, $$2$$ for a Classical and $$6$$ for a Mathematical scholarship,then find No. of ways these scholarships can be awarded is,
  • $$48$$
  • $$12$$
  • $$24$$
  • $$8$$
Given:   $$\dfrac {20!}{18!}=380$$
  • True
  • False
  • Either
  • Neither
In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential.
  • True
  • False
  • Either
  • Neither
$$\sum_{r=0}^{n} r^{2} . ^{n}C_{r}p^{r}q^{n-r} ,  where \  \ p + q = 1,$$ is simplified to:
  • $$npq + n^{2}p^{2}$$
  • $$n^{2}p^{2}q^{2} + np$$
  • $$np(p + q)$$
  • $$\frac{p(q+1)}{2}$$
Which among the following is/are correct?
  • If an operations can be performed in 'm' different ways and a second operation can be performed in 'n' different ways, then both of these operations can be performed in $$'m\times n'$$ ways together.
  • The number of arrangements of n different objects taken all at a times is n!.
  • The number of permutations of n different things taken r at a time, when each thing may be repeated any number of times is n.
  • The number of circular permutations of 'n' different things taken all at a time is $$\frac {1}{2}(n-1)!$$, if clockwise and anticlockwise orders are taken as different.
Which among the following is/are not correct?
  • $$^6C_2+ \ ^6C_1= \ ^7C_2$$
  • $$^6C_1+ \ ^6C_2= \ ^6C_2$$
  • $$^6C_2+ \ ^6C_1= \ ^7C_1$$
  • $$^6C_2- \ ^6C_1= \ ^7C_2$$
Two series of a question booklet for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical series side by side and that the students sitting one behind the other should have the same series?
  • $$2 \times ^{12}C_6 \times(6!)^2$$
  • 6! x 6!
  • 7! x 7!
  • 2 x 6!
How many different signals can be transmitted by arranging 3 red, 2 yellow and 2 green flags on a pole? [Assume that all the 7 flags are used to transmit a signal].
  • 210
  • 215
  • 220
  • 225
A five digit number divisible by $$3$$ is to be formed using the numerals $$0, 1, 2, 3, 4$$ and $$5$$ without repetition. The total number of ways in which this can be done is:
  • $$211$$
  • $$216$$
  • $$221$$
  • $$311$$
How many strings of letters can possibly by formed using the above rules such that the third letter of the string is e?
  • 8
  • 9
  • 10
  • 11
How may strings of letters can possibly be formed using the above rules?
  • 40
  • 45
  • 30
  • 35
How many numbers greater than a million can be formed with the digits $$2, 3, 0, 3, 4, 2, 3$$?
  • $$360$$
  • $$366$$
  • $$356$$
  • $$370$$
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