Permutations & Combinations · NDA previous year questions

Consider the following statements: 1. Value of $\sin \theta$ oscillates between $-1$ and 1. 2. Value of $\cos \theta$ oscillates between 0 and 1. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Consider the following statements: 1. $n\left({\sin }^{2}67{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }\right)>1$ for all positive integers $n\ge 2$. 2. If $x$ is any positive real number, then $nx>1$ for all positive integers $n\ge 2$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

In how many ways can the letters of the word 'GLOOMY' be arranged so that the two O's should not be together ?

1. A.240
2. B.480
3. C.600
4. D.720

The number of permutations that can be formed from all the letters of the word 'BASEBALL' is:

1. A.540
2. B.1260
3. C.3780
4. D.5040

Consider the following statements : 1. If $f(x)=x^3$ and $g(y)=y^3$ then f = g. 2. Identity function is not always a bijection. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the number of ways that 4 boys and 3 girls can be seated so that boys and girls alternate?

1. A.12
2. B.72
3. C.120
4. D.144

If the difference between the roots of $a{x}^2+bx+c=0$ is 1, then which one of the following is correct?

1. A.$b^2=a(a+4c)$
2. B.$a^2=b(b+4c)$
3. C.$a^2=c(a+4c)$
4. D.$b^2=a(b+4c)$

There are 4 candidates for the post of a lecturer in Mathematics and one is to be selected by votes of 5 men. What is the number of ways in which the votes can be given?

1. A.1048
2. B.1072
3. C.1024
4. D.625

What is the value of ${\sum }_{r=1}^n\frac{P(n,r)}{r!}$?

1. A.$2^n-1$
2. B.$2^n$
3. C.$2^{n-1}$
4. D.$2^n+1$

What are the co-ordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$?

1. A.$(2,9)$
2. B.$(5,6)$
3. C.$(-5,6)$
4. D.$(6,5)$

In how many ways can 6 girls be seated in 2 empty chairs?

1. A.10
2. B.15
3. C.24
4. D.30

What is the value of ${\log }_{10}\left(\frac{9}{8}\right)-{\log }_{10}\left(\frac{27}{32}\right)+{\log }_{10}\left(\frac{3}{4}\right)$?

1. A.3
2. B.2
3. C.1
4. D.0

What is the total number of combinations of $n$ different things taken $1,2,3,\dots ,n$ at a time?

1. A.$2^{n+1}$
2. B.$2^{2n+1}$
3. C.$2^{n-1}$
4. D.$2^n-1$

5 books are to be chosen from a lot of 10 books. If $m$ is the number of ways of choice when one specified book is always included and $n$ is the number of ways of choice when a specified book is always excluded, then which one of the following is correct?

1. A.$m>n$
2. B.$m=n$
3. C.$m=n-1$
4. D.$m=n-2$

What is the number of words that can be formed from the letters of the word 'UNIVERSAL', the vowels remaining always together?

1. A.720
2. B.1440
3. C.17280
4. D.21540

In how many ways can a committee consisting of $3$ men and $2$ women be formed from $7$ men and $5$ women?

1. A.$45$
2. B.$350$
3. C.$700$
4. D.$4200$

If $\vec{a}=\hat{i}-\hat{k},\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k},\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$, then $\vec{a}\cdot \left(\vec{b}\times \vec{c}\right)$ depends on

1. A.$x$ only
2. B.$y$ only
3. C.both $x$ and $y$
4. D.neither $x$ nor $y$

A team of 8 players is to be chosen from a group of 12 players. Out of the eight players one is to be elected as captain and another as vice-captain. In how many ways can this be done?

1. A.27720
2. B.13860
3. C.6930
4. D.495

What is the number of ways of arranging the letters of the word 'BANANA' so that no two N's appear together?

1. A.40
2. B.60
3. C.80
4. D.100

In how many ways can the letters of the word 'GARDEN' be arranged so that in each word the vowels should appear in alphabetical order?

1. A.120
2. B.240
3. C.360
4. D.480

If $V=\{x:x+2=0\}$, $R=\{x:x^2+2x=0\}$ and $S=\{x:x^2+x-2=0\}$, then for what value of $x$ is $V=R=S$?

1. A.$0$
2. B.$-1$
3. C.$-2$
4. D.$1$

Out of a group of 20 teachers in a school, 10 teach Mathematics, 9 teach Physics and 7 teach Chemistry. 4 teach both Mathematics and Physics, but none teach both Mathematics and Chemistry. What is the number of teachers who teach both Chemistry and Physics?

1. A.1
2. B.2
3. C.3
4. D.4