Suppose m men and n women are to be seated in a row so that no two
women sit together. If m > n, show that the number of ways in which they can be seated is\(m!(m+1)! \over (m-n+1)!\)

 

(i) How many numbers are there between 99 and 1000 having 7 in the units place?
(ii) How many numbers are there between 99 and 1000 having atleast one of their
digits 7?

In how many ways can 5 children be arranged in a line such that (i) two
particular children of them are always together (ii) two particular children of them are never together

If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the 49th word?

In how many ways 3 mathematics books, 4 history books, 3 chemistry books and 2 biology books can be arranged on a shelf so that all books of the same subjects are together

A student has to answer 10 questions, choosing atleast 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how  many ways can the student choose 10 questions?

Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.

 In a small village, there are 87 families, of which 52 families have atmost 2 children. In a rural development programme 20 families are to be chosen for assistance, of which atleast 18 families must have at most 2 children. In how many ways can the choice be made?

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

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