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

The number of factors (excluding $$1$$ and the expression itself) of the product of $$a^7b^4c^3def$$ where $$a,b,c,d,e,f$$ are all prime numbers is 
  • $$1278$$
  • $$1360$$
  • $$1100$$
  • $$1005$$
If $$\space ^{n+1}C_{r+1}:  ^nC_r :  ^{n-1}C_{r-1} = 11:6:3$$, then find the values of $$n$$ and $$r$$.
  • $$n=10,\space r=5$$
  • $$n=9,\space r=4$$
  • $$n=11,\space r=5$$
  • $$n=10,\space r=4$$
There are three papers of $$100$$ marks each in an examination. In how many ways can a student get $$150$$ marks such that he gets at least $$60%$$ in two papers?
  • $$1480$$
  • $$1488$$
  • $$1520$$
  • $$1526$$
A person always prefers to eat parantha and vegetable dish in his meal. How many ways can he make his plate in a marriage party if there are three types of paranthas, four types of vegetable dishes, three types of salads, and two types of sauces?
  • $$3360$$
  • $$4096$$
  • $$3000$$
  • None of these
Evaluate 
$$^{47}C_4 + \displaystyle\sum_{j=0}^{3}{^{50-j}C_3} + \sum_{k=0}^{5}{^{56-k}C_{53-k}}$$
  • $$^{57}C_4$$
  • $$^{57}C_5$$
  • $$^{56}C_6$$
  • $$^{56}C_5$$
For $$2\le r \le n$$, $$\left(\begin{matrix}n \\ r\end{matrix}\right) + 2\left(\begin{matrix}n \\ r-1 \end{matrix}\right) + \left(\begin{matrix}n \\ r - 2 \end{matrix}\right) \space =$$
  • $$\left(\begin{matrix}n + 1 \\ r - 1\end{matrix}\right)$$
  • $$2\left(\begin{matrix}n + 1 \\ r + 1\end{matrix}\right)$$
  • $$2\left(\begin{matrix}n + 2 \\ r \end{matrix}\right)$$
  • $$\left(\begin{matrix}n + 2 \\ r \end{matrix}\right)$$
For any positive integer $$m,\space n \space(\mbox{ with } n\ge m) = ^nC_m$$. 
$$\left(\begin{matrix}n \\ m\end{matrix}\right) + \left(\begin{matrix}n - 1 \\ m\end{matrix}\right) + \left(\begin{matrix}n - 2 \\ m\end{matrix}\right) + ... + \left(\begin{matrix}m \\ m\end{matrix}\right) = $$
  • $$\left(\begin{matrix}n+1 \\ m+1\end{matrix}\right)$$
  • $$\left(\begin{matrix}n\\ m+1\end{matrix}\right)$$
  • $$\left(\begin{matrix}n\\ m\end{matrix}\right)$$
  • $$\left(\begin{matrix}n-1\\ m\end{matrix}\right)$$
The number of positive integers satisfying the inequality
$$\quad ^{n+1}C_{n-2} - ^{n+1}C_{n-1} \le 100$$ is
  • One
  • Eight
  • Five
  • None of these
The straight lines $$I_1, I_2, I_3$$ are parallel and lie in the same plane. A total number of $$m$$ points on $$I_1$$; $$n$$ points on $$I_2$$; $$k$$ points on $$I_3$$, the maximum number of triangles formed with vertices at these points are
  • $$^{m+n+k}C_3$$
  • $$^{m+n+k}C_3 - ^mC_3 - ^nC_3 - ^kC_3$$
  • $$^mC_3 + ^nC_3 + ^kC_3$$
  • None of these
If $$r, s$$ and $$t$$ are prime numbers and $$p, q$$ are positive integers such that the LCM of $$p,q$$ is $$\displaystyle r^{2}t^{4}s^{2}$$ then the number of ordered pair $$(p, q)$$ is 
  • 254
  • 252
  • 225
  • 224
If $$\displaystyle ^{7}C_{r} + 3 ^{7}C_{r+1} + 3 ^{7}C_{r+2} + ^{7}C_{r+3} > ^{10}C_{4}$$, then the quadratic equation whose roots are $$\displaystyle \alpha, \: \beta$$ and $$\displaystyle \alpha^{r-1}, \: \beta^{r-1}$$ have
  • no common roots
  • only one common root
  • two common root
  • none of these

In the figure,two 4-digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4-digit number and in no place the addition is with carrying, is
134574_0a1bd3eb3f364e778bf559df70f3b609.png
  • $$\displaystyle 55^4$$
  • 220
  • $$\displaystyle 45^4$$
  • none of these
If $$^nC_{r-1}=36, ^nC_r=84$$ and $$^nC_{r+1}=126$$, then r is
  • $$1$$
  • $$2$$
  • $$3$$
  • none of these
The number of words of four letters containing equal number of vowels and consonants, repetition being allowed, is 
  • $$\displaystyle 105^2$$
  • $$\displaystyle 210 \: \times \: 243$$
  • $$\displaystyle 105 \: \times \: 243$$
  • none of these
The value of $$\displaystyle ^{40}C_{31}+\sum_{j=0}^{10} \: ^{40+j}C_{10+j}$$ is equal to
  • $$\displaystyle ^{51}C_{20}$$
  • $$\displaystyle 2 . ^{50}C_{20}$$
  • $$2 . ^{45}C_{15}$$
  • none of these
The number of different matrices that can be formed with elements 0,1,2 or 3, each matrix having 4 elements, is
  • $$\displaystyle 3\times 2^4$$
  • $$\displaystyle 2 \times 4^4$$
  • $$\displaystyle 3 \times 4^4$$
  • none of these
The value of $$\displaystyle \sum_{r=1}^{n}r(^{n}C_{r}+^{r}P_{r})$$is
  • $$\displaystyle n\cdot 2^{n-1}-1$$
  • $$\displaystyle n\cdot 2^{n-1}+(n+1)!$$
  • $$\displaystyle n\cdot 2^{n-1}+(n+1)!-1$$
  • $$\displaystyle n^{2}+n+5$$
The  number of signals that can be given using any number of flags of 5 different colors, is 
  • $$225$$
  • $$325$$
  • $$215$$
  • $$315$$
The value of the expression $$ ^{47}C_{4}+\sum_{f= 0}^{6}\ ^{52-f}C_{3} $$ equals 
  • $$ ^{47}C_{5} $$
  • $$ ^{52}C_{5} $$
  • $$ ^{52}C_{4} $$
  • $$ ^{52}C_{3} $$
The exponent of 7 in the coefficient of the greatest term in the expansion of  $$\displaystyle (1+x)^{200}$$ is
  • 0
  • 1
  • 2
  • 3
The mean value of $$^{20}C_0,\dfrac{^{20}C_2}{3},\dfrac{^{20}C_4}{5},\cdots ,\dfrac{^{20}C_{20}}{21}$$ equals
  • $$\dfrac{2^{20}}{21}$$
  • $$\dfrac{2^{19}}{21}$$
  • $$\dfrac{2^{20}}{3\times 77}$$
  • $$\dfrac{2^{19}}{33\times 7}$$
If $$\displaystyle \frac{^{n}C_{r}+3^{n}C_{r+1}+3^{n}C_{r+2}+^{n}C_{r+3}}{^{n}C_{r}+4^{n}C_{r+1}+6^{n}C_{r+2}+4^{n}C_{r+3}+^{n}C_{r+4}}=\frac{r+k}{n+k}$$, then the value of $$k$$ equals 
  • $$1$$
  • $$2$$
  • $$4$$
  • None of these
The expression $$ ^{n+4}C_{r}-^{n}C_{r}-3.^{n}C_{r-1}-3^{n}C_{r-2}-^{n}C_{r-3} $$
  • $$ ^{n+3}C_{r-1} $$
  • $$ ^{n+2}C_{r+1} $$
  • $$ ^{n+4}C_{r+1} $$
  • $$ ^{n+1}C_{r-1} $$
$$\displaystyle \sum _{ 0\le i\le  }^{  }{ \sum _{ j\le 10 }^{  }{ ^{ 10 }{ { C }_{ j } }^{ j }{ { C }_{ i } } }  } $$ is equal to
  • $$3^{10}$$
  • $$3^{10}-1$$
  • $$2^{10}$$
  • $$2^{10}-1$$
The sum $$\displaystyle \sum_{i=0}^{m}\binom{10}{i}\binom{20}{m-i}$$ be maximum when m is
  • 15
  • 5
  • 10
  • 20
The value of the expression 
$$\displaystyle2^{k}\binom{n}{0}\binom{n}{k}-2^{k-1}\binom{n}{1}\binom{n-1}{k-1}+2^{k-2}\binom{n}{2}\binom{n-2}{k-2}..+(-1)^{k}\binom{n}{k}\binom{n-k}{0}$$ is
  • $$\displaystyle \binom{n}{k}$$
  • $$\displaystyle \binom{n+1}{k}$$
  • $$\displaystyle \binom{n+1}{k+1}$$
  • $$\displaystyle \binom{n-1}{k-1}$$
If $$\displaystyle ^{100}C_{3} = 161700$$, then $$^{100}C_{97}$$ is equal to___.
  • 53,900
  • 40,425
  • 1,61,700
  • 16,17,000
A road network as shown in the figure connect four cities. In how many ways can you start from any city (say A) and come back to it without travelling on the same road more than once ?
264017_3bbc4030bc7046e098b8425af44cc9d3.png
  • 8
  • 12
  • 9
  • 16
If $$\displaystyle ^{n}C_{3}=  ^{n}C_{5'}$$ then find the value of n:
  • $$9$$
  • $$10$$
  • $$8$$
  • $$7$$
If $$ { _{  }^{ n }{ C } }_{ 4 },{ _{  }^{ n }{ C } }_{ 5 }$$ and $$ { _{  }^{ n }{ C } }_{ 6 }$$ are in AP, then $$n$$ is
  • $$7$$ or $$14$$
  • $$7$$
  • $$14$$
  • $$14$$ or $$21$$
If $$^{19}C_r$$ and $$^{19}C_{r-1}$$ are in the ratio 2:3, then find $$^{14}C_r$$
  • 91
  • 81
  • 71
  • 61
A bag contains n white and n black balls. Pairs of balls are drawn at random without replacement successively, until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is 14,then n =
  • 6
  • 5
  • 4
  • 3
Three persons entered a railway compartment in which $$5$$ seats were vacant. Find the number of ways in which they can be seated
  • $$30$$
  • $$45$$
  • $$120$$
  • $$60$$
How many $$3$$ digit numbers can we make using the digits $$4,5,7$$ and $$9$$ and where repetition is allowed?
  • $${3}^{4}$$
  • $${3}^{3}$$
  • $${4}^{4}$$
  • $$12$$
  • $${4}^{3}$$
A password for a computer system requires exactly $$6$$ characters. Each character can be either one of the $$26$$ letters from A to Z or one of the ten digits from $$0$$ to $$9$$. The first character must be a letter and the last character must be a digit. How many different possible passwords are there?
  • Less than $$10^{7}$$
  • Between $$10^{7}$$ and $$10^{8}$$
  • Between $$10^{8}$$ and $$10^{9}$$
  • Between $$10^{9}$$ and $$10^{10}$$
  • More than $$10^{10}$$
The exponent of $$5$$ in $$^{120}C_{60}$$, is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$3$$
If $$^nC_{r-1}=36, ^nC_r = 84$$ and $$^nC_{r+1} = 126$$ then the value of $$^nC_8$$ is:
  • $$10$$
  • $$7$$
  • $$9$$
  • $$8$$
The value of the expression $$^{47}C_4$$ +$$ \sum_{j=1}^{5} { }^{52-j}C_3$$ is
  • $$^{51}C_4$$
  • $$^{52}C_4$$
  • $$^{52}C_3$$
  • $$^{53}C_4$$
If $$^{n-1}C_r = (k^2 - 3) ^nC_{r+1} ,$$ then k belongs to the interval
  • $$[\sqrt{-3}, \sqrt3 ]$$
  • $$(-\infty,-2)$$
  • $$(2, \infty)$$
  • $$(\sqrt3, 2]$$
If $$^{20}C_r = ^{20}C_{r-10}$$, then the value of $$^{18}C_r$$ is:
  • $$4896$$
  • $$5432$$
  • $$816$$
  • $$1632$$
The exponent of $$7$$ in $$^{120}C_{50}$$,  is
  • $$0$$
  • $$2$$
  • $$4$$
  • None of these
For a chess tournament $$13$$ people were selected for quarter finals. Each person plays two matches with the other. How many matches have been held in the whole tournament?
  • $$144$$
  • $$156$$
  • $$185$$
  • $$116$$
The area of regular polygon of n sides with length of side $$\sqrt{3}$$, where n > 1 and satisfies the relation $$\displaystyle \sum_{r=0}^{n} \frac{n^2-3n+3}{2\ ^nC_r}=\sum_{r=0}^n\frac{r}{^nC_r}$$ is :
  • $$30\sqrt{3}$$
  • $$\dfrac{3\sqrt{3}}{4}$$
  • $$\dfrac{5\sqrt{3}}{4}$$
  • $$\dfrac{3\sqrt{3}}{2}$$
$$12$$ people came for a carroms tournament. All were divided into pairs of $$2$$ each. After all the matches if it found that half of the selected pairs had to play three matches to decide the winner in a best of three process, how many matches have been held?
 
  • $$165$$
  • $$185$$
  • $$198$$
  • $$132$$
If $$^{19}C_{3r} = ^{19}C_{r+3} $$, then r is equal to:
  • $$5$$
  • $$4$$
  • $$3$$
  • $$2$$
How many alphabets need to be there in a language if one were to make $$1$$ million distinct $$3$$ digit initials using the alphabets of the language?
  • $$10$$
  • $$100$$
  • $$56$$
  • $$26$$
 After every get-together every person present shakes the hand of every other person. If there were 105 handshakes in all, how many persons were present in the party?
  • $$16$$
  • $$15$$
  • $$13$$
  • $$14$$
$$8064$$ is resolved into all possible product of two factors. Find the number of ways in which this can be done?
  • $$24$$
  • $$21$$
  • $$20$$
  • None of these
How many straight lines can be formed from $$11$$ points in a plane out of which no three points are collinear?
  • $$66$$
  • $$50$$
  • $$55$$
  • $$60$$
A college offers $$7$$ courses in the morning and $$5$$ in the evening. Find possible number of choices with the student who want to study one course in the morning and one in the evening.
  • $$35$$
  • $$12$$
  • $$49$$
  • $$25$$
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