Combination with Restrictions

Q. A student is to answer 10 out of 13 questions in an examination such that he must choose at least four from the first five questions. The number of choices available to him is

1. 196
2. 280
3. 140
4. 346

Q. Let the product of all the divisors of 1440 be P. If P is divisible by ${24}^x$, then the maximum value of x is

1. 32
2. 36
3. 28
4. 30

Q.

Question 1 (b)
Observe the patterns of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators.

b)

If the number of digits formed is taken to be 'n', the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern. How many segments are required to form 5, 10, 100 digits of the kind

Q. The number of different six digit numbers, the sum of whose digits is odd is

1. 
2. 
3. 
4. 

Q. If a = $2^3\times 3^3\times 7^2$and $b=2^2\times 3^2\times 5^3,$ which is true?

1. $HCF=2^2\times 3^2;LCM=2^3\times 3^3\times 5^3\times 7^2$
   
2. $HCF=2^3\times 3^2;LCM=2^3\times 3^2\times 5^2\times 7^3$
   
3. $HCF=2^3\times 3^2;LCM=2^3\times 3^2\times 5^3\times 7^3$
   
4. $LCM=2^2\times 3^2;HCF=2^3\times 3^3\times 5^3\times 7^2$

Q.

p = $2^2$.$3^2$.$q^2$ (where q is a prime $<$ 7) is the prime factorisation representation of ‘p’. What is the value of p?

1. p = 600

2. Can’t say

3. p = 36

4. p = 900

5. p = 700

Q. Find the number of cubes in given figure and give the top, front, left side and right side view (arrow indicating the front view).

Q. Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then, the value of $\frac{m}{n}$ ?___

Q. The number of words that can be formed with the letters of the word DAUGHTER so that vowels remain always together is

1. 4320
2. 1320
3. 2320
4. 3320

Q. Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A\times B$ having 3 or more elements is

1. 256
2. 220
3. 219
4. 211

Q. A man bought two pens costing $3.26$ and $4.82$ respectively. Find the total money spent by him using estimation.

1. $8$
2. $9$
3. $7$
4. $5$

Q. Total $5$ -digit numbers divisible by $6$ can be formed using $0,1,2,3,4,5$ if repetition of digits is allowed is:

1. $108$
2. $4680$
3. $60$
4. $48$

Q. The number of diagonals that can be drawn from one vertex in a 7-sided polygon is ____.

1. 4
2. 5
3. 6
4. 7

Q. The number of $3-$ digit numbers can be formed from the digits $1,2,3,4$ and $5$ assuming repetition of digits not allowed is

1. $20$
2. $30$
3. $40$
4. $60$

Q. If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is $r^2{t}^4{s}^2$ then the number of ordered pair (p, q) is

1. 252
2. 254
3. 225
4. 224

Q. In how many ways $4$ prizes can be given to $3$ boys, if all boy is eligible for all the prizes

1. $81$
2. $256$
3. $12$
4. None of these

Q. A rectangle with sides 2m -1 and 2n -1 is divided into squares of unit length by drawing parallel lines as shows in the diagram, then the number of rectangles possible with odd side lengths is

1. $m^2{n}^2$
2. $m(m+1)n(n+1)$
3. $4^{m+n-1}$
4. $(m+n-1{)}^2$

Q.

Suppose $2016$ points of the circumference of a circle points are coloured red and the remaining points are coloured blue. Find the minimum possible value of a natural number $n$, for which there exists a regular $n$- sided polygon whose all vertices are blue.

1. $3$
2. $5$
3. $6$
4. Not possible

Q. Find the number of cubes in given figure and give the top, front, left side and right side view (arrow indicating the front view).

Q. Consider the set of all positive integers n for which $f\left(n\right)=n!+\left(n+1\right)!+\left(n+2\right)!$ is divisible by $49$.

1. The number of integers n in (1, 15) is 3
2. The number of integers n in (5, 17) is 4
3. The number of integers n in (1, 20) is 8
4. The number of integer n in (1, 20) is 9

Q. $D\subset A,D\ne B$

Q. The sides AB, BC, and CA of a triangle ABC have 3, 4, and 5 interior points, respectively, on them. The number of triangles that can be constructed using these interior points as vertices is ___.

Q. Colour the multiples of $2$ and $5$ in the grid.
The common multiples from the grid are

1. $10,20,30$
2. $10,20$
3. $2,10$
4. $5,10$

Q. Number greater than or equal to 1000 but less than or equal to 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed) is

1. 125
2. 105
3. 375
4. 625

Q. $16x^2+40x+K$ is a perfect square polynomial, then k =

Q. Geometrically expand the polynomial: $(t^2+u^2+2v^2{)}^2$

Q.

Three cities of our big world are connected as shown in the image. The resulting figure looks somewhat similar to a triangle. These kinds of figures are studied in –

1. Both A and B

2. Euclidean Geometry

3. Don’t know

4. Non - Euclidean Geometry

Q.

Out of 18 points in a plane, no three are in the same straight line except 5 points which are collinear. How many straight lines can be formed by joining them?

1. 288

2. 143
   

3. 145
   

4. 144
   

Q. In each of the following, determine whether the statements is true or false if it is true prove it if it false given an example.
If $x\in A$ and $A\subset ̸B$ then $x\in B$

1. True
2. False

Q. If all the words that are formed by permuting the letters of the word ${}^{\prime }PLAT{E}^{\prime }$ are arranged in alphabetical order, then the rank of the word ${}^{\prime }PLAT{E}^{\prime }$ is

1. $81$
2. $86$
3. $80$
4. $85$