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

The number of 3-digit numbers, with distinct digits, that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ and divisible by 3 is
(1) 80
(2) 120
(3) 40
(4) 108

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) $5 : 3$

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

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$, is:
(1) 820
(2) 780
(3) 901
(4) 861

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

$n$ - 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) 6
(2) 8
(3) 9
(4) 7

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

Total number of 6-digit numbers in which only and all the five digits $1, 3, 5, 7$ and 9 appears, is
(1) $\frac { 1 } { 2 } (6!)$
(2) $6!$
(3) $5 ^ { 6 }$
(4) $\frac { 5 } { 2 } (6!)$

If the number of five digit numbers with distinct digits and 2 at the $10 ^ { \text {th} }$ place is $336 k$, then $k$ is equal to:
(1) 4
(2) 6
(3) 7
(4) 8

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ?
(1) $2 ! 3 ! 4$ !
(2) $( 3 ! ) ^ { 3 } \cdot ( 4 ! )$
(3) $( 3 ! ) 2 . ( 4 ! )$
(4) $3 ! ( 4 ! ) ^ { 3 }$

The value of $\left( 2 \cdot { } ^ { 1 } P _ { 0 } - 3 \cdot { } ^ { 2 } P _ { 1 } + 4 \cdot { } ^ { 3 } P _ { 2 } - \right.$ up to $51 ^ { \text {th} }$ term $) + ( 1 ! - 2 ! + 3 ! -$ up to $51 ^ { \text {th} }$ term) is equal to
(1) $1 - 51 ( 51 )$ !
(2) $1 + ( 51 )$ !
(3) $1 + ( 52 )$ !
(4) 1

The sum of all the 4-digit distinct numbers that can be formed with the digits $1, 2, 2$ and 3 is:
(1) 26664
(2) 122664
(3) 122234
(4) 22264

The total number of positive integral solutions $( x , y , z )$ such that $x y z = 24$ is :
(1) 45
(2) 30
(3) 36
(4) 24

If ${ } ^ { n } P _ { r } = { } ^ { n } P _ { r + 1 }$ and ${ } ^ { n } C _ { r } = { } ^ { n } C _ { r - 1 }$, then the value of $r$ is equal to:
(1) 1
(2) 4
(3) 2
(4) 3