All the letters of the word PUBLIC are written in all possible orders and these words are written

as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

(A) 580                   (B) 582                  (C) 576                  (D) 578

The number of arrangements of the letters of the word

"INDEPENDENCE" in which all the vowels always occur together is

(A)  14800                    (B) 16800               (C) 18000                  (D) 33600

The number of ways in which 5 girls and 7 boys can be seated at a

round table so that no two girls sit together, is

(A) $7{\left(360\right)}^2$             (B) $126{(5!)}^2$                (C) $7{\left(720\right)}^2$              (D) $720$

If the number of words, with or without meaning. which can be made using all the letters of the word

MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to

(A) 5670                   (B) 945                 (C) 2835                   (D) 1890

Eight persons are to be transported from city A to city B in three cars of different makes. If each car

can accommodate at most three persons, then the number of ways, in which they can be transported, is

(A) 3360                    (B) 1120                   (C) 1680                    (D) 560

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a

dictionary with serial numbers, then the serial number of the word THAMS is

(A) 103                    (B) 104                     (C) 101                      (D) 102

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed

using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to

(A) 132                       (B) 120                   (C) 72                    (D) 96

All words, with or without meaning, are made using all the letters of the word MONDAY.

These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is

(A) 327                     (B) 328                      (C) 326                     (D) 324

The total number of three-digit numbers, divisible by 3, which can be formed using the

digits 1, 3, 5, 8, if repetition of digits is allowed, is

(A) 18                        (B) 21                    (C) 22                        (D) 20

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the

elements in each row is 1 and the sum of all the elements in each column is also 1, is

(A) 225                      (B) 120                 (C) 125                     (D) 150

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is

(A) 168                  (B) 120                     (C) 220                   (D) 48

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is

(A) 72                    (B) 120                   (C) 6                       (D) 12

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series.

Then the serial number of the word TOUGH is

(A) 86                      (B) 89                    (C) 79                    (D) 84

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

(A) 400                    (B) 507                 (C) 432                  (D) 472

The number of ways of selecting two numbers $a$ and $b,a\in \{2,4,6...,100\}$ 

and $b\in \{1,3,5...,99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is

(A) 54                      (B) 108                 (C) 268                      (D) 186

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ________ .

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _________ .

In an examination, 5 students have been allotted their seats as per their roll numbers.

The number of ways, in which none of the students sits on the allotted seat, is __________ .

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ .

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________ .

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits.Then the maximum number of trials necessary to obtain the correct code is _____ .

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ___________ .

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is __________ .

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ .

The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is __________ .

Let $x$ and $y$ be distinct integers where $1\le x\le 25$ and $1\le y\le 25.$Then, the number of ways of choosing $x$ and $y,$ such that $x+y$ is divisible by 5, is _________ .

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is _______ .

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _________ .

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ .

Let $\sum _{n=0}^{\infty }\frac{n^3\left(\right(2n)!)+(2n-1)(n!)}{(n!)\left(\right(2n)!)}=ae+\frac{b}{e}+c,$ where $a,b,c\in \mathrm{\mathbb{Z}}$ and $e=\sum _{n=0}^{\infty }\frac{1}{n!}$. Then $a^2-b+c$ is equal to ________ .

The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _______ .

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _______ .

If ${}^{2n+1}{P}_{n-1}:{}^{2n-1}{P}_n=11:21,$ then $n^2+n+15$ is equal to ______ .

If ${}^{2n}{C}_3:{}^n{C}_3=10:1,$ then the ratio $(n^2+3n):(n^2-3n+4)$ is

(A) 27 : 11                   (B) 65 : 37                (C) 2 : 1               (D) 35 : 16

Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A x B each having at least 3 and at most 6 elements is

(A) 752                      (B) 792                  (C) 772                     (D) 782

The number of triplets (x, y, z), where x, y, z are distinct non negative integers satisfying x + y + z = 15, is

(A) 80                        (B) 114                      (C) 92                          (D) 136

$\sum _{k=0}^6{}^{51-k}{C}_3$ is equal to

(A) ${}^{51}{C}_3-{}^{45}{C}_3$                (B) ${}^{52}{C}_4-{}^{45}{C}_4$               (C) ${}^{52}{C}_3-{}^{45}{C}_3$                (D) ${}^{51}{C}_4-{}^{45}{C}_4$

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is __________ .

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ___________ .

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is ___________ .

Number of integral solution to the equation x + y + z = 21, where $x\ge 1, y\ge 3, z\ge 4$, is equal to __________ .

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is _______ .

Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is ________ .

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _______ .

If all the six digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0<x_1<x_2<x_3<x_4<x_5<x_6$ are arranged in the increasing order, then the sum of the digits in the 72th number is ___________ .