Practice Exercises

The number of employees in a nationalized bank in a small town is $10$, out of which $4$are females and the rest are males. A committee of $5$is to be formed. If $m$ be the number of ways to form such a committee in which there is at least one female employee and $n$ be the number of ways to form such a committee which includes at least $2$male employees, then find the ratio $m:n$.

A question paper is split into two parts, part A and part B. Part A contains $5$questions and part B has $4$questions. Each question in part A has an alternative. A student has to attempt at least one question from each part. Find the number of ways in which the student can attempt the question paper?

How many integers between $1$and $1000000$ have the sum of their digits equal to $18$?

A person invites his $10$friends for a party and places them $5$at one round table, and $5$on the other round table. Find the number of ways in each case in which he can arrange the guests.

A person invites his $10$friends for a party. Find the number of ways in which he can arrange $4$at one round table and $6$at the other round table.

In how many ways $4$boys and $4$girls can be seated in a row so that boys and girls are alternate?

There is a vertical stack of books marked $1, 2$ and $3$ on table A, with $1$ at the bottom and $3$ on the top. These are to be placed vertically on table B with $1$ at the bottom and $2$ on the top, by making a series of moves from one table to the other. During a move, the topmost book, or the topmost two books, or all the three, can be moved from one of the tables to the other. If there are any books on the other table, the stack being transferred should be placed on to; of the existing books, without changing the order of books in the stack that is being moved in that move. If there are no books on the other table, the stack is simply placed on the other table without distributing the order of books in it. What is the minimum number of moves in which the above task can be accomplished? 

A shipping clerk has five boxes of different but unknown weights each weighing less than $100 \mathrm{kg}$. The clerk weight the boxes in pairs. The weights obtained are $110,112,113,114,115,116,117$, $118,120$ and $121 \mathrm{kg}$. What is the weight of the heaviest box?