CBSE Questions for Class 11 Engineering Maths Permutations And Combinations Quiz 5 - MCQExams.com

$$\displaystyle \begin{pmatrix} 35 \\ 6 \end{pmatrix}+\sum_{r=0}^{10}\begin{pmatrix} 45-r \\ 5 \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}$$
then $$x-y$$ is equal to:
  • $$39$$
  • $$29$$
  • $$52$$
  • $$40$$
$$\displaystyle \begin{pmatrix} 47 \\ 4 \end{pmatrix}$$+$$\displaystyle \sum_{j=1}^{5}$$ $$\displaystyle \begin{pmatrix} 52-j \\ 3 \end{pmatrix}$$=$$\displaystyle \begin{pmatrix} x \\ y \end{pmatrix}$$ then $$\displaystyle \frac{x}{y}$$ =
  • 11
  • 12
  • 13
  • 14
If $$^8C_r = ^8C_{r+2}$$, then the value of $$^rC_2$$ is:
  • 8
  • 3
  • 5
  • 2
$$^{50}C_{11}+^{50}C_{12}+^{51}C_{13}-^{52}C_{13}=$$
  • $$^{52}C_{14}$$
  • $$0$$
  • $$^{53}C_{13}$$
  • none of these
The value of  $$^{95}C_4+\displaystyle \sum_{j=1}^5 {\;}^{100-j}C_3$$ is
  • $$^{95}C_5$$
  • $$^{100}C_4$$
  • $$^{99}C_4$$
  • $$^{100}C_5$$
If $$^nC_3 + ^nC_4 > ^{n+1}C_3$$, then
  • $$n > 6$$
  • $$n > 7$$
  • $$n < 6$$
  • none of these
If $$'n'$$ is an integer between $$0$$ and $$21$$, then the minimum value of $$n!\left( 21-n \right) !$$ is
  • $$9!2!$$
  • $$10!11!$$
  • $$20!$$
  • $$21!$$
$$\displaystyle \sum_{r=0}^m {\;}^{n+r}C_n=$$
  • $$^{n+m+1}C_{n+1}$$
  • $$^{n+m+2}C_{n}$$
  • $$^{n+m+3}C_{n+1}$$
  • none of these
If $$2\times$$ $$^nC_5 = 9\times$$ $$^{n-2}C_5$$, then the value of n will be:
  • 7
  • 10
  • 9
  • 5
If $$^{15}C_{3r}=^{15}C_{r+3}$$, then the value of r is:
  • 3
  • 4
  • 5
  • 8
Let $$A=(x|x$$ is a prime number and $$x<300$$ > the number of different rational numbers, whose numerator and denominator belong to $$A$$ is:
  • 91
  • 84
  • 106
  • None of these
If n and r are two positive integers such that $$n\geq r$$, then $$^nC_{r+1} + ^nC_r =$$
  • $$^nC_{n-r}$$
  • $$^nC_r $$
  • $$^{n-1}C_r $$
  • $$^{n+1}C_r+1 $$
If $$(^{15}C_r + ^{15}C_{r-1}) (^{15}C_{15-r} + ^{15}C_{16-r}) = (^{16}C_{13})^2$$, then the value of $$r$$ is
  • $$r=2$$
  • $$r=3$$
  • $$r=4$$
  • none of these
In a game called 'odd man out', $$ m (m > 2)$$ persons toss a coin to determine who will buy refreshment for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. The probability that there is a loser in any game is
  • $$1/2 m$$
  • $$m/2 ^{m-1}$$
  • $$2/m$$
  • none of these
Three players play a total of $$9$$ games. In each game, one person wins and the other two lose; the winner gets $$2$$ points and the losers lose $$1$$ each. The number of ways in which they can play all the $$9$$ games and finish each with a zero score is
  • $$84$$
  • $$1680$$
  • $$7056$$
  • $$0$$
Let $$ S= \left\{1,2,3,.......n\right\} $$ and $$ A =\left\{(a, b) \left. \right | 1 \geq a, b \geq n\right\} = S \times S\: $$. A subset $$ B$$ of $$A $$ is said to be a good subset if $$ (x, x) \in B$$ for every $$x \in S.$$ Then the number of good subsets of $$A$$ is
  • $$1$$
  • $$2^{n}$$
  • $$2^{n(n-1)}$$
  • $$2^{n^{2}}$$
If $$4.^nC_6 = 33.^{n-3}C_3$$ then $$n$$ is equal to
  • $$9$$
  • $$10$$
  • $$11$$
  • none of these
There are 8 buses running from Kota to Jaipur and 10 buses running from Jaipur to Delhi. In how many ways a Person can travel from Kota to Delhi via Jaipur by bus?
  • 80
  • 8
  • 10
  • 160
Forty teams play a tournament. Each of them plays with every other team just once. Each game result is a win for one team. If each team has a $$50\%$$ chance of winnings each game, the number of ways such that at the end of the tournament, every team has won a different number of games is:
  • $$\dfrac{1}{780}$$
  • $$40!2^{780}$$
  • $$40! 3^{780}$$
  • None of these
One mapping is selected at randon from all mappings of the set $$ S = \left\{ 1,2,3, ....., n \right\}$$ into itself. If the probability that mapping is one-one is $$3/32$$ then the value of $$n$$ is
  • $$2$$
  • $$3$$
  • $$4$$
  • None of these
The sum of the series $$\displaystyle \sum_{r=0}^{10} {\;}^{20}C_r$$ is
  • $$2^{20}$$
  • $$2^{19}$$
  • $$2^{19}+\displaystyle \frac {1}{2}^{20}C_{10}$$
  • $$2^{19}-\displaystyle \frac {1}{2}^{20}C_{10}$$
If $$\displaystyle^{n}P_{r}=  360$$  and  $$^ {n}C_{r}=15$$ then find the value of r
  • $$5$$
  • $$4$$
  • $$3$$
  • $$2$$
The value of $$\displaystyle ^{ 19 }C_{ 18 }+^{ 19 }C_{ 17 }$$
  • $$1200$$
  • $$2000$$
  • $$190$$
  • None of these
IF$$\displaystyle ^{n}C_{r}=^{n}P_{r}$$  then r can be____
  • 0
  • 1
  • 3
  • Either(1) or (2)
There are $$5$$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back, is
  • $$25$$
  • $$20$$
  • $$10$$
  • $$5$$
If $$(1+x)^n=C_0+C_1x+C_2x^2+ ......+C_nx^n,$$ then the value of $$C_0 + 2C_1+ 3 C_2 + ...... (n + 1) C_n$$ will be
  • $$(n + 2)2^{n - 1}$$
  • $$(n + 1)2^{n }$$
  • $$(n -1)2^{n - 1}$$
  • $$(n + 2)2^{n}$$
Find the value of $$\displaystyle ^{10}C_{10}$$
  • $$2$$
  • $$1$$
  • $$4$$
  • $$6$$
In how many ways can 8 books be distributed among 5 students if each student is eligible for any number of books?
  • $$40$$
  • $$\displaystyle 5^{8}$$
  • $$360$$
  • $$\displaystyle 8^{5}$$
$$\displaystyle ^{5}C_{4}=$$___
  • $$4$$
  • $$5$$
  • $$20$$
  • $$24$$
Different calenders for the month of February are made so as to serve for all the coming years. The number of such calenders is
  • $$7$$
  • $$2$$
  • $$14$$
  • None of these
In $$\displaystyle ^{3}C_{r}$$ the value of r can be ___
  • $$2$$
  • $$4$$
  • $$6$$
  • All of these
A three-digit code for certain locks uses the digits $$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$$ according to the following constraints. The first digit cannot be $$0$$ or $$1$$, the second digit must be $$0$$ or $$1$$, and the second and third digits cannot both be $$0$$ in the same code. How many different codes are possible?
  • $$144$$
  • $$152$$
  • $$160$$
  • $$168$$
  • $$176$$
In how many ways can 3 diamond cards be drawn simultaneously from a pack of cards?
  • $$78$$
  • $$1716$$
  • $$286$$
  • $$13$$
A researcher plans to identify each participant in a $$174$$ certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are $$12$$ participants,and each participant is to receive a different code?
  • $$4$$
  • $$5$$
  • $$6$$
  • $$7$$
  • $$8$$
The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?
434550.png
  • $$6$$
  • $$7$$
  • $$12$$
  • $$14$$
  • $$17$$
Mario's Pizza has 2 choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers every pizza in extra cheese as well as regular. If Linda's volleyball team decides to order a pizza with 4 toppings, how many different choices do the teammates have at Mario's Pizza?
  • $$24$$
  • $$32$$
  • $$28$$
  • $$20$$
From the consecutive integers -10 to 10, inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers? 
  • $$\displaystyle { \left( -10 \right) }^{ 20 }$$
  • $$\displaystyle { \left( -10 \right) }^{ 10 }$$
  • $$\displaystyle 0$$
  • $$\displaystyle { -\left( 10 \right) }^{ 19 }$$
  • $$\displaystyle { -\left( 10 \right) }^{ 20 }$$
If $$\displaystyle ^{ 16 }{ C }_{ r }=^{ 16 }{ C }_{ r+1 }$$, then the value of $$\displaystyle ^{ r }{ P }_{ r-3 }$$ is
$$\displaystyle $$
  • 31
  • 120
  • 210
  • 840
  • No option is correct
Four speakers will address a meeting where speaker $$Q$$ will always speak after $$P$$. Then, the number of ways in which the order of speakers can be prepared is
  • $$256$$
  • $$128$$
  • $$24$$
  • $$12$$
If $$_{  }^{ n }{ { P }_{ r } }=30240$$ and $$_{  }^{ n }{ { C }_{ r } }=252$$, then the ordered pair $$\left( n,r \right) $$ is equal to
  • $$\left( 12,6 \right) $$
  • $$\left( 10,5 \right) $$
  • $$\left( 9,4 \right) $$
  • $$\left( 16,7 \right) $$
If $$^{n}C_{12} = ^{n}C_{6}$$, then $$^{n}C_{2}$$is equal to:
  • $$72$$
  • $$153$$
  • $$306$$
  • $$2556$$
$$^n C_{r-1}\, =\, 330,\, ^nC_r\, =\, 462,\, ^nC_{r+1}\, =\, 462\, \Rightarrow\, r\, =$$
  • 3
  • 4
  • 5
  • 6
A vehicle registration number consists of $$2$$ letters of English alphabet followed by $$4$$ digits, where the first digit is not zero. Then, the total number of vehicles with distinct registration numbers is
  • $$26^{2} \times 10^{4}$$
  • $$^{26}P_{2} \times\ ^{10}P_{4}$$
  • $$^{26}P_{2} \times 9\times\ ^{10}P_{3}$$
  • $$26^{2} \times 9\times 10^{3}$$
If $$a=99^{50}+100^{50}$$ and $$b=101^{50}$$, then :
  • $$ a < b$$
  • $$a=b$$
  • $$a > b$$
  • $$a-b=100^{49}$$
If n is an integer with $$0\le n \le 11$$ then the minimum value of $$n!(11-n)!$$ is attained when a value of n = 
  • 11
  • 5
  • 7
  • 9
Kathy is scheduling the first four periods of her school day. She needs to fill those periods with calculus, art, literature, and physics, and each of these courses is offered during each of the first four periods. Calculate the total number of different schedules Kathy can choose from.
  • $$1$$
  • $$4$$
  • $$12$$
  • $$24$$
  • $$120$$
If $$^{(n-1)}C_3+^{(n-1)}C_4 > ^nC_3$$, then the minimum value of $$n$$ is:
  • 5
  • 6
  • 7
  • 8
The number of words that can be formed out of the letters of the word "ARTICLE" so that the vowels occupy even places is
  • $$574$$
  • $$36$$
  • $$754$$
  • $$144$$
If k is odd then $$\displaystyle ^{ k }{ C }_{ r }$$, is maximum for r equal to
  • $$\displaystyle \frac { 1 }{ 2 } \left( k-1 \right) $$
  • $$\displaystyle \frac { 1 }{ 2 } \left( k+1 \right) $$
  • $$\displaystyle k-1$$
  • $$\displaystyle k$$
If $$X, Y, Z, D, E$$ and $$F$$ are $$6$$ distinct points on the circumference of a circle, find the number of different chords which can be drawn using any $$2$$ of the $$6$$ points.
  • $$6$$
  • $$12$$
  • $$15$$
  • $$30$$
  • $$36$$
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