Circular Permutation

Q. $5$ boys and $5$ girls had to sit alternately around a round table. This can be done in $N$ ways. Then the value of $N$ is

Q. A round table conference is to be held among $20$ delegates belonging from $20$ different countries. The number of ways they can be seated

1. without any restriction is $19!$
2. when there is atleast two persons between two particular delegates is $18\times 18!$
3. when two particular delegates should never sit opposite to each other is $19!$
4. when two particular delegates should always sit together is $2\times 18!$

Q. The number of ways that 7 boys can be seated round a table so that 2 particular boys are always separated, is

1. 960
2. 720
3. 480
4. 240

Q. There are $2$ brothers among a group of $20$ persons. The number of ways the group can be arranged around a circle so that there is exactly one person between the two brothers is

1. $2\times 17!$
2. $18!\times 18$
3. $2\times 18!$
4. $2\times 17!\times 17$

Q. Six persons $A,B,C,D,E,F$ are to be seated at a circular table. If $A$ should have either $B$ or $C$ on his immediate right and $B$ must always have either $C$ or $D$ on his immediate right. Then the total number of possible arrangements is

Q.

The number of ways in which a necklace can be formed by using $5$ identical red beads and $6$ identical black beads is?

1. $\frac{11!}{6!\times 4!}$

2. ${}^{11}{P}_6$

3. $\frac{10!}{2\left(6!\times 5!\right)}$

4. None of these

Q.

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish for sit on one particular side and two on the other side. In how many ways can they be seated ?

Q. $20$ persons are sitting in a particular arrangement around a circular table.Number of ways of selection of three persons from them such that no two were sitting adjacent to each other is

Q. The total number of ways in which $6$ persons can be seated at a round table, so that all persons shall not have the same neighbours in any two arrangements.

1. $60$
2. $120$
3. $180$
4. $360$

Q. In a playground, $3$ sisters and $5$ other girls are playing together. The number of ways in which all the girls be seated in a circular order so that the three sisters are not seated together is

1. $5040$
2. $4920$
3. $4320$
4. $2160$

Q.

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

Q.

Write the number of ways in which 7 men and 7 women can sit on a round table such that no two women sit together.

Q. The total number of ways in which $3$ girls and $3$ boys be seated at a round table, so that any $2$ and only $2$ of the girls are always together is

Q. Number of ways in which $10$ different diamonds can be arranged to make a necklace is

1. $\frac{9!}{2}$
2. $10!$
3. $9!$
4. $\frac{10!}{2}$

Q. Three boys and three girls are to be seated around a circular table. Among them, the boy $X$ does not want any girl neighbour and the girls $Y$ does not want any boy neighbour. Then the number of possible arrangements is

1. $4$
2. $6$
3. $8$
4. $10$

Q.

If eleven members of a committee sit at a round table so that the President and Secretary always sit together, then the number of arrangements is?

1. $10!\times 2$

2. $10!$

3. $9!\times 2$

4. None of these

Q. Twelve persons are to be arranged around two round tables such that one table can accommodate seven persons and another table can accommodate five persons only.

The total number of possible arrangements if two particular persons $A$ and $B$ want to be together is

1. ${}^{10}{C}_7\times 6!\times 3!\times 2!+{}^{10}{C}_5\times 4!\times 5!\times 2!$
2. ${}^{10}{C}_5\times 6!\times 3!+{}^{10}{C}_7\times 4!\times 5!$
3. ${}^{10}{C}_7\times 6!\times 2!+{}^{10}{C}_5\times 5!\times 2!$
4. ${}^{10}{C}_5\times 6!\times 3!\times 2!+{}^{10}{C}_7\times 4!\times 5!$

Q.

A normal window has the shape of a rectangle surmounted by a semicircle. (thus the diameter of the semicircle is equal to the width of the rectangle.) if the perimeter of the window is $16\mathrm{ft}$, find the value of $x$ so that the greatest possible amount of light is admitted. (give your answer correct to two decimal places.)

Q. If $20$ persons are invited for a party then number of different ways the persons and the host can be seated in a circular table, if two particular persons are to be seated on either side of the host is

1. $2\times 18!$
2. $20!$
3. $18!$
4. $3!\times 18!$

Q. If $8$ persons $A,B,C,D,E,F,G,H$ are to be seated around a circular table, then the number of possible arrangements

1. without any restriction is $7!$
2. when $A$ and $B$ should be seated together is $6!$
3. when $A$ and $B$ should be seated opposite to each other is $6!$
4. when there should be exactly two persons between $A$ and $B$ is $720$

Q. The number of ways in which 6 Boys and 5 Girls can sit in a row so that no two girls and no two boys are together is

1. 
2. 
   
3. 2(5! 6!)
4. 5! 6!

Q. A team of $8$ persons including ''Captain'' and ''Vice-captain'' are to be seated around a circular table, then the number of possible arrangements

1. without any restriction is $7!$
2. when ''Captain'' and ''Vice-captain'' should be seated opposite to each other is $5!$
3. when there should be atleast one person between ''Captain'' and ''Vice-captain'' is $5\times 6!$
4. when there should be exactly one person between ''Captain'' and ''Vice-captain'' is $12\times 5!$

Q. there are 5 men and 5 ladies to dine at a round table. In how many ways they can seat themselves so that no two ladies are together?

Q.

In how many ways can five keys be put in a ring?

1. $\frac{1}{2}4!$

2. $\frac{1}{2}5!$

3. 4!

4. 5!

Q. There are $6$ boys and $5$ girls. There are two round tables with $7$ chairs and $4$ chairs.

1. Number of ways of arranging all of them is $\frac{11!}{28}$
2. Number of ways of arranging all of them so that all girls are at same table is $\frac{(6!{)}^2}{8}$
3. Number of ways of arranging all of them so that all boys are at same table is $\frac{5!6!}{4}$
4. Number of ways of arranging all of them so that all boys are at one table or all girls are at one table is $6!\left(120\right)$

Q. The total number of ways in which $20$ different pearls of two colours can be set alternately on a necklace, there being $10$ pearls of each colour, is

1. $11!$
2. $4\times (8!{)}^2$
3. $6\times (9!{)}^2$
4. $5\times (9!{)}^2$

Q. There are $2n$ guests at a dinner party. Supposing that the master and mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be placed next to one another, the number of ways in which the company can be placed is

1. $(4n^2-6n+4)\times (n-1)!$
2. $(2n^2-3n+2)\times (2n-2)!$
3. $(4n^2-6n+4)\times (2n-2)!$
4. $(4n^2-6n+4)\times (2n-3)!$

Q. There are $10$ persons sitting around a circular table from which three persons are selected for the board of directors. The number of ways to select them such that no two persons are adjacent to each other is

Q. How many garlands can be formed using 10 different flowers?

1. $\frac{9!}{2}$
2. $\frac{10!}{2}$
3. $10!$
4. $10!\times 2!$

Q. 5 girls and 5 boys are to be seated around a circular table such that no two girls are together. Find the number of ways to be seated.