Combination with Restrictions

Q.

The number of selecting at least $4$ candidates from $8$ candidates is

1. $270$

2. $70$

3. $163$

4. None of these

Q. The digits of a three-digit positive integer are in A.P. and their sum is $15$. The number obtained by reversing the digits is $594$ less than the original number. Then the unit place of the number is

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

Q. You are given $8$ balls of different colours (black, white, ...). The number of ways in which these balls can be arranged in a row so that the two balls of particular colour (say red and white) may never come together, is

1. $8!-2\times 7!$
2. $6\times 7!$
3. $2\times 6!\times {}^7{C}_2$
4. $2(7!\times 8!)$

Q. A committee of $5$ men and $3$ women is to be formed out of $7$ men and $6$ women. If two particular women are not to be included together in the committee, then the number of committees that can be formed is

1. $420$
2. $540$
3. $336$
4. $216$

Q. The number of all $3\times 3$ matrices $A$, with entries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $\left(A{A}^T\right)$ is $3$, is

Q. A total of $5$ different mathematics books, $4$ different physics books and $2$ different chemistry books are to be arranged in a row in a book shelf. Which of the following is (are) TRUE?

1. The number of arrangements in which two chemistry books are separated is $9\times 10!$
2. The number of arrangements in which four physics books are together is $8!4!$
3. The number of arrangements in which no two mathematics books are together is $(7\cdot 6)(6!)$
4. The number of arrangements in which the books of the same subject are all together is $12(4!\cdot 5!)$

Q. If $4$ integers are to be selected from {$1,2,3,......20$} such that the sum of the integers should be the multiple of $4$, then number of ways to select the numbers are

1. $1220$
2. $1120$
3. $1200$
4. $970$

Q. We have to choose $11$ players for cricket team from $8$ batsmen, $6$ bowlers, $4$ all rounders and $2$ wicket keepers. Number of selections when a particular batsman and a particular wicket keeper do not want to play together, is

1. $2^{18}{C}_{10}$
2. ${}^{19}{C}_{11}{+}^{18}{C}_{10}$
3. ${}^{19}{C}_{10}{+}^{19}{C}_{11}$
4. None of these

Q. An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at most three of them are red is

Q. The number of non - congruent rectangles that can be found on a chess board is

Q. Consider three boxes, each containing $10$ balls labelled $1,2,\dots ,10.$ Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i,$ the label of the ball drawn from the $i^{\text{th}}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $n_1<n_2<n_3$ is :

1. $120$
2. $164$
3. $82$
4. $240$

Q. The number of five letter words containing 3 vowels and 2 consonants that can be formed using the letters of the word EQUATION so that two consonants occur together is

1. ${}^5{C}_3{}^3{C}_{5}5!$
2. ${}^5{C}_3{}^3{C}_{5}2!3!$
3. ${}^5{P}_3{}^3{P}_{2}5!$
4. ${}^5{C}_3{}^3{C}_{2}2!4!$

Q. The total number of $6$ digit numbers that can be formed, having the property that every succeeding digit is greater than the preceding digit is equal to

1. ${}^9{C}_3$
2. ${}^{10}{C}_3$
3. ${}^9{P}_3$
4. ${}^{10}{P}_3$

Q. Determine the average of all four digit numbers that can be made using all the digits $2,3,5,7$ and $8$ exactly once?

1. $3993$
2. $5555$
3. $5486$
4. $5347$

Q. The number of selections of $6$ different letters that can be made from the words $\text{NISHIT}$ and $\text{RAHUL}$ so that each selection consists of $3$ letters from each word, is

Q. The number of ways in which $13$ non-distinguishable books can be distributed among $7$ students so that every student get at least one book and at least one student gets $4$ books but not more, is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $341$
2. $351$
3. $343$
4. $371$

Q. Five balls are to be placed in three boxes. Each box can hold all the five balls so that no box remains empty.
If balls and boxes are identical then number of ways is

Q. The number of $4$-digit natural number which contains

1. Exactly two distinct digits is $567$
2. At least two distinct digits is $8990$
3. At most two distinct digits is $577$
4. At least three identical digits is $333$.

Q. Kanchan has $10$ friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is

1. $2\times {}^8{C}_3$
2. $2\times {}^8{C}_4$
3. ${}^8{C}_3$
4. ${}^8{C}_4$

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. 28
2. 30
3. 32
4. 36

Q. The kindergarten teacher has $25$ kids in her class. She takes $5$ of them at a time, to zoological garden as often as she can, without taking the same $5$ kids more than once. Then the number of visits, the teacher makes to the garden exceeds that of a kid by

1. ${}^{25}{C}_5-{}^{24}{C}_4$
2. ${}^{24}{C}_5$
3. ${}^{25}{C}_5-{}^{24}{C}_5$
4. ${}^{24}{C}_4$

Q. The number of ways in which a team of $11$ players be formed out of $25$ players, if $6$ out of them are always to be included and $5$ always to be excluded is

Q. Let $n$ be four digit positive integer in which all the digits are different. If $x$ is number of odd integers and $y$ is number of even integers, then

1. $x<y$
2. $x>y$
3. $x+y=4500$
4. $|x-y|=56$

Q. The number of five letter words containing 3 vowels and 2 consonants that can be formed using the letters of the word EQUATION so that two consonants occur together is

1. ${}^5{C}_3{}^3{C}_{5}5!$
2. ${}^5{C}_3{}^3{C}_{5}2!3!$
3. ${}^5{P}_3{}^3{P}_{2}5!$
4. ${}^5{C}_3{}^3{C}_{2}2!4!$

Q. If $10$ different balls has to placed in $4$ distinct boxes at random, then the probability that two of these boxes contain exactly $2$ and $3$ balls is :

1. $\frac{965}{2^{10}}$
2. $\frac{945}{2^{10}}$
3. $\frac{945}{2^{11}}$
4. $\frac{965}{2^{11}}$

Q. If the sides $AB,BC$ and $CA$of a triangle $ABC$ have $3,5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices is equal to:

1. $360$
2. $240$
3. $364$
4. $333$

Q. Consider the letters of the word $MATHEMATICS.$ Then the possible number of words

1. without any restriction is $\frac{11!}{212!2!}$
2. when all the repeating letters are at odd position is $5!\times \frac{6!}{2!2!2!}$
3. when the word starts with consonant is $\frac{7\times 10!}{8}$
4. having $7$ letters where all letters are unique is${}^8{P}_7$

Q. Determine the average of all four digit numbers that can be made using all the digits $2,3,5,7$ and $8$ exactly once?

1. $3993$
2. $5555$
3. $5486$
4. $5347$

Q. Consider the set $S=\{0,1,2,3,\cdots ,9\}.$ Let $m$ denotes the number of ways the two numbers $a,b$ with replacement chosen from $S$ such that $|a-b|$ $>6$ and $n$ denotes the number of $10$ digit numbers that can be formed using each and every digit of $S,$ which are divisible by $2$ but not by $3,$ then which of the following is/are correct?

1. $m+n=12$
2. $m+n=16$
3. $m-n=12$
4. $m-n=8$

Q. Two straight line intersect at a point $O.$ Points $A_1,A_2,...A_n$ are taken on a line and points $B_1,B_2,...B_n$ are taken on the other line. If the point $O$ is not to be used, then number of triangles that can be drawn using these points as vertices, is

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