Permutation and Combination

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

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.

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? 

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.

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? 

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

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?

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 $m:n$.

Six persons $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E},$ and $\mathrm{F}$are to be seated at a circular table. In how many ways can this be done if $\mathrm{A}$must always have either $\mathrm{B}$or $\mathrm{C}$on his immediate right and $\mathrm{B}$ must always have either $\mathrm{C}$ or $\mathrm{D}$ on his immediate right?

Number of ways in which $n$ distinct things can be distributed among $n$ persons so that at least one person does not get anything is $232$. Find $n$.

How many positive integers less than $1000$are $6$times the sum of their digits?

In how many ways can one arrange letters the in the word 'INSTITUTION' such that no two same letters comes together along with the following conditions:

(A) There is no ' $\mathrm{T}$ ' which is immediately preceded as well as followed by ' $\mathrm{N}$ '.

(B) There is no ' $\mathrm{T}$ ' which is immediately preceded as well as followed by ' $\mathrm{T}$ '.

If $\mathrm{N}$ be an element of the set $\mathrm{A}=\{1, 2, 3, 5, 6, 10, 15, 30\}$, and $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are integers, such that $\mathrm{PQR}=\mathrm{N}$, then the number of positive integral solutions of $\mathrm{PQR}=\mathrm{N}$ is:

Let, ${\left(1+x+x^2\right)}^9={\alpha }_0+{\alpha }_{1}x+\dots +{\alpha }_{18}{x}^{18}$, then which of the following is true?

The streets of a city are arranged like the lines of a chessboard. There are $m$ streets running north-south and $n$ streets running in east-west direction. What is the number of ways in which a man can travel on these streets from the north-west to the south-east corner, going by the shortest possible distance?

In a test of $10$multiple choice questions of one correct answer, each having $4$alternative answers, then the number of ways to put ticks at random for the answers to all the questions is:

If the number of ways of selecting $\mathrm{k}$ coupons out of an unlimited number of coupons bearing the letters $\mathrm{A},\mathrm{T}$, and $\mathrm{C}$ so that they cannot be used to spell the word $\mathrm{CAT}$ is $93$, then what is the value of $\mathrm{k}$ ?

Find the number of non-negative integer solutions to the system of equations $a+b+c+d+e=20$ and $a+b+c=5$.

In a chess tournament, every person played one game with every other person in the group. The total number of games that men played between themselves exceeded those played by men with women by $18$. If there were $4$women in the tournament, then in total, how many games were played in the tournament?

Consider $\mathrm{S}=(1,2,3,\dots ,10)$. In how many ways two numbers from $\mathrm{S}$ can be selected so that the sum of the numbers selected is a double-digit number?