There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student $S _ { 1 }$ gets the previously allotted seat $R _ { 1 }$, and NONE of the remaining students gets the seat previously allotted to him/her is
(A) $\frac { 3 } { 40 }$
(B) $\frac { 1 } { 8 }$
(C) $\frac { 7 } { 40 }$
(D) $\frac { 1 } { 5 }$

There are five students $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$ and $S _ { 5 }$ in a music class and for them there are five seats $R _ { 1 } , R _ { 2 } , R _ { 3 } , R _ { 4 }$ and $R _ { 5 }$ arranged in a row, where initially the seat $R _ { i }$ is allotted to the student $S _ { i } , i = 1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats. For $i = 1,2,3,4$, let $T _ { i }$ denote the event that the students $S _ { i }$ and $S _ { i + 1 }$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _ { 1 } \cap T _ { 2 } \cap T _ { 3 } \cap T _ { 4 }$ is
(A) $\frac { 1 } { 15 }$
(B) $\frac { 1 } { 10 }$
(C) $\frac { 7 } { 60 }$
(D) $\frac { 1 } { 5 }$

Five persons $A$, $B$, $C$, $D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is\_\_\_\_

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$ is $\_\_\_\_$.

Let the set of all relations $R$ on the set $\{ a , b , c , d , e , f \}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal { S }$.
Then the number of elements in $\mathcal { S }$ is $\_\_\_\_$ .

Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits 0 and 1 appears exactly twice in $x$, is equal to $\_\_\_\_$ .

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is
(1) $6 ! 7 !$
(2) $( 6 ! ) ^ { 2 }$
(3) $( 7 ! ) ^ { 2 }$
(4) $7 !$

The number of arrangements that can be formed from the letters $a , b , c , d , e , f$ taken 3 at a time without repetition and each arrangement containing at least one vowel, is
(1) 96
(2) 128
(3) 24
(4) 72

Statement 1: If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^{p}$. Statement 2: The total number of selections of $p$ different objects out of $q$ objects is ${}^{q}C_{p}$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.

5-digit numbers are to be formed using $2,3,5,7,9$ without repeating the digits. If $p$ be the number of such numbers that exceed 20000 and $q$ be the number of those that lie between 30000 and 90000, then $p : q$ is:
(1) $6 : 5$
(2) $3 : 2$
(3) $4 : 3$
(4) $5 : 3$

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval
(1) $( 11,13 ]$
(2) $( 14,17 )$
(3) $[ 10,12 )$
(4) $[ 8,9 ]$

The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers $3,4,5$ and $6$, without repetition is:
(1) 18
(2) 36
(3) 108
(4) 432

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:
(1) 216
(2) 192
(3) 120
(4) 72

The number of integers greater than 6000 that can be formed, using the digits $3,5,6,7$ and 8 , without repetition is
(1) 72
(2) 216
(3) 192
(4) 120

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is:
(1) 46th
(2) 59th
(3) 52nd
(4) 58th

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is: (1) 46th (2) 59th (3) 52nd (4) 58th

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:
(1) $47 ^ { t h }$
(2) $45 ^ { t h }$
(3) $46 ^ { t h }$
(4) $44 ^ { \text {th } }$

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:
(1) At least 750 but less than 1000
(2) At least 1000
(3) Less than 500
(4) At least 500 but less than 750

The number of numbers between 2,000 and 5,000 that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of 3 is
(1) 36
(2) 30
(3) 24
(4) 48

$n$-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of $n$ for which 900 such distinct numbers can be formed is :
(1) 9
(2) 7
(3) 8
(4) 6

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84 , then the value of $m$ is :
(1) 11
(2) 12
(3) 7
(4) 9

All possible numbers are formed using the digits $1,1,2,2,2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is
(1) 175
(2) 162
(3) 180
(4) 160

The number of natural numbers less than 7000 which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to:
(1) 375
(2) 250
(3) 374
(4) 372

The number of four-digit numbers strictly greater than 4321 that can be formed using the digit $0,1,2,3,4,5$ (repetition of digits is allowed) is:
(1) 360
(2) 288
(3) 306
(4) 310

Let $Z$ be the set of integers. If $A = \left\{ x \in Z : 2 ^ { ( x + 2 ) \left( x ^ { 2 } - 5 x + 6 \right) } = 1 \right\}$ and $B = \{ x \in Z : - 3 < 2 x - 1 < 9 \}$, then the number of subsets of the set $A \times B$, is :
(1) $2 ^ { 12 }$
(2) $2 ^ { 10 }$
(3) $2 ^ { 18 }$
(4) $2 ^ { 15 }$