Find the number of permutations of n different things taken r at a time
such that two specific things occur together.

There are four bus routes between A and B; and three bus routes
between B and C. A man can travel round-trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?
(A) 72
(B) 144
(C) 14
(D) 19

In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men and 5 women?
(A) 45
(B) 350
(C) 4200
(D) 230

All the letters of the word ‘EAMCOT’ are arranged in different possible
ways. The number of such arrangements in which no two vowels are adjacent to each  other is
(A) 360
(B) 144
(C) 72
(D) 54

 Ten different letters of alphabet are given. Words with five letters are
formed from these given letters. Then the number of words which have atleast one letter repeated is
(A) 69760
(B) 30240
(C) 99748
(D) 99784

The number of signals that can be sent by 6 flags of different colours
taking one or more at a time is
(A) 63
(B) 1956
(C) 720
(D) 21

In an examination there are three multiple choice questions and each
question has 4 choices. Number of ways in which a student can fail to get all answer correct is
(A) 11
(B) 12
(C) 27
(D) 63

Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4  and then men select from the remaining chairs. Find the total number of possible
arrangements.

If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary. Then what is the rank of the word RACHIT?

A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. Find the number of different ways of doing questions.

Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the point.