1 . 
From amongst 36 teachers in a school, one principal and one viceprincipal are to be appointed. In how many ways can this be done ? 

Answer & Explanation
Answer : Option
A


Explanation : 

Principal can be appointed in 36 ways. Vice principal can be appointed in the remaining 35 ways. Total number of ways = 36 $\times$ 35 = 1260 





2 . 
A boy has 3 library cards and 8 books of his interest in the library. Of these 8, he does not want to borrow chemistry part II unless Chemistry part I is also borrowed. In how many ways can he choose the three books to be borrowed ? 

Answer & Explanation
Answer : Option
D


Explanation : 







3 . 
In how many ways can six different rings be worn on four fingers of one hand ? 

Answer & Explanation
Answer : Option
C


Explanation : 







4 . 
In how many ways can 7 persons be seated at a round table if 2 particular persons must not sit next to each other ? 

Answer & Explanation
Answer : Option
C


Explanation : 

Total no. of unrestricted arrangements = (7  1) ! = 6 ! When two particular person always sit together, the total no. of arrangements = 6!  2 $\times$ 5! Required no. of arrangements = 6!  2 $\times$ 5! = 5! (6  2) = 5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 4 = 480. 





5 . 
In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together ? 

Answer & Explanation
Answer : Option
C


Explanation : 







6 . 
In an examination paper there are two sections each containing 4 questions. A candidate is required to attempt 5 questions but not more than 3 questions from any particular section. In how many ways can 5 questions be selected ? 

Answer & Explanation
Answer : Option
B


Explanation : 

Under the given restrictions, 5 questions can be selected in the following ways : 2 questions from the first section and 3 questions from the second section 





7 . 
There are 4 candidates for the post of a lecturer in Mathematics and one is to be selected by votes of 5 men. The number of ways in which the votes can be given is 

Answer & Explanation
Answer : Option
D


Explanation : 







8 . 
The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by 

Answer & Explanation
Answer : Option
A


Explanation : 







9 . 
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is 

Answer & Explanation
Answer : Option
C


Explanation : 







10 . 
There are 100 students in a college class of which 36 are boys studying statistics and 13 girls not studying statistics. If there are 55 girls in all, then the probability that a boy picked up at random is not studying statistics, is 

Answer & Explanation
Answer : Option
C


Explanation : 

There are 55 girls and 45 boys in the college. Out of 45 boys, 36 are studying Statistics and 9 are not studying statistics. The probability that a boy picked up at random is not studying Statistics = $9 \over 45$ = $1 \over 5$ 





11 . 
The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by : 

Answer & Explanation
Answer : Option
A


Explanation : 

No. of ways in which 6 men and 5 women can dine at a round table = 6! $\times$ 5! 





12 . 
The number of ways in which a team of eleven players can be selected from 22 players including 2 of them and excluding 4 of them is: 

Answer & Explanation
Answer : Option
C


Explanation : 







13 . 
In how many different ways can the letters of the word 'PRETTY' be arranged? 

Answer & Explanation
Answer : Option
C


Explanation : 







14 . 
Three boys and three girls are to be seated around a table in a circle. Among them the boy X does not want any girl neighbour and the girl Y does not want any boy neighbour. How many such arrangements are possible ? 

Answer & Explanation
Answer : Option
C


Explanation : 

Four possible arrangements are : 





15 . 
Answer these questions on the basis of the information given below :
From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many different ways can it be done so that the committee has at least one woman? 

Answer & Explanation
Answer : Option
C


Explanation : 







16 . 
Answer these questions on the basis of the information given below :
From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many different ways can it be done, so that the committee has at least 2 men? 

Answer & Explanation
Answer : Option
D


Explanation : 







17 . 
Answer these questions on the basis of the information given below :
From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many ways can 5 persons be chosen from 6 boys and 4 girls so as to include exactly one girl? 

Answer & Explanation
Answer : Option
E


Explanation : 







18 . 
Answer these questions on the basis of the information given below :
From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many different ways can the letters of the word CORPORATION be arranged? 

Answer & Explanation
Answer : Option
A


Explanation : 

CORPORATION= 11 letters 'O' comes thrice, 'R' twice. total no. of ways = $11! \over 3! 2!$ = 3326400 





19 . 
In how many different ways can the letters of the word 'COUNTRY' be arranged in such a way that the vowels always come together? 

Answer & Explanation
Answer : Option
B


Explanation : 

There are seven letters in the word 'COUNTRY' and two vowels O and U. Considering two vowels as one unit, total number of letters will be 5 + 1 = 6. So, number of arrangements = 6! Now, the two vowels can be arranged in 2! ways among themselves. Total number of ways = 6! $\times$ 2! = 1440 





20 . 
In how many different ways can the letters of the word 'PROBLEM' be arranged ? 

Answer & Explanation
Answer : Option
C


Explanation : 

The word PROBLEM consists of 7 distinct letters. Number of arrangements = 7! = 7 $\times$ 6 $\times$ 5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 1 = 5040 





21 . 
Out of eight crew members three particular members can sit only on the left side. Another two particular members can sit only on the right side. Find the number of ways in which the crew can be arranged so that four men can sit on each side. 

Answer & Explanation
Answer : Option
D


Explanation : 







22 . 
In how many different ways can the letters of the word 'OFFICES' be arranged? 

Answer & Explanation
Answer : Option
A


Explanation : 







23 . 
In how many different ways can the letters of the word 'ARMOUR' be arranged? 

Answer & Explanation
Answer : Option
E


Explanation : 







24 . 
There are six teachers. Out of them, two are primary teachers and two are secondary teachers. They are to stand in a row, so as the primary teachers, middle teachers and secondary teachers are always in a set. The number of ways in which they can do so, is 

Answer & Explanation
Answer : Option
B


Explanation : 







25 . 
In how many different ways can 4 boys and 3 girls be arranged in a row such that all boys stand together and all the girls stand together? 

Answer & Explanation
Answer : Option
C


Explanation : 

Total number of ways to stand boys and girls together = 4! $\times$ 3! $\times$ 2! = 4 $\times$ 3 $\times$ 2 $\times$ 3 $\times$ 2 $\times$ 2 = 288 





26 . 
In how many ways can a committee of 4 people be chosen out of 8 people ? 

Answer & Explanation
Answer : Option
C


Explanation : 

Total number of ways = ${8_C}_4$ = 70 





27 . 
From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done? 

Answer & Explanation
Answer : Option
D


Explanation : 







28 . 
In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together? 

Answer & Explanation
Answer : Option
C


Explanation : 

The word 'LEADING' has 7 different letters. When the vowels EAI are always together, they can be supposed to form one letter. Then, we have to arrange the letters LNDG (EAI). Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways. The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 $\times$ 6) = 720. 





29 . 
How many 3digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated? 

Answer & Explanation
Answer : Option
D


Explanation : 

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it. The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place. The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it. Required number of numbers = (1 $\times$ 5 $\times$ 4) = 20. 





30 . 
How many ways can 4 prizes be given away to 3 boys, if each boy is eligible for all the prizes? 

Answer & Explanation
Answer : Option
C


Explanation : 

Any one prize can be given to any one of the 3 boys and hence there are 3 ways of distributing each prize. Hence, the 4 prizes can be distributed in 34= 81 ways 





