Circular Permutations

The number of ways, in which $5$ girls and $7$ boys can be seated at a round  table so that no two girls sit together is

$7$ boys and $5$ girls are to be seated around a circular table such that no two girls sit together is 

In how many ways can 12 gentlemen sit around a round table so that three specified gentlemen are always together?

The number of ways in which $5$ red beads and $4$ yellow beads of different sizes can be made out to form a necklace so that no two yellow beads come together is

Number of necklaces of $8$ beads each can be made from $15$ beads of various colours is

Garlands are formed using $6$ red roses and $6$ yellow roses of different sizes. The number of arrangements in garland which have red roses and yellow roses come alternatively is

There are $n$ seats round a table numbered $1,2,3,....,n$. The number of ways in which $m\left(\le n\right)$ persons can take seats is

In how many ways can $7$ person sit around a round table so that all shall not have same neighbours in any two arrangements?

The number of ways in which $6$ men and $5$ women can dine at a round table, if no two women are to sit together is

$5$ boys $\left(B_1, B_2, B_3, B_4\right.$ and $\left.B_5\right)$ and $5$ girls $\left(G_1,G_2,G_3,G_4\right.$ and $\left.G_5\right)$ are to be seated around a round table such that boy and girl sit alternately and $B_1$ does not sit beside $G_i\forall i\in \left\{1,2,3,4,5\right\}.$ If the number of such arrangements is $N$, then the sum of digits of $N$ is equal to

Number of circular arrangement of $11$ identical green balls, $1$ red, $1$ white and $1$ blue ball is

Number of ways to seat $8$ men and $8$ women around a round table where one particular man and one particular woman always sit together, and men and women alternate is equal to

In a circle $20$ person are sitted, then the number of way of selecting $5$ person such that no two person are consecutive, are :-

The number of ways in which $5$ boys and $4$ girls can be arranged on a circular table such that no two girls sit together and two particular boys are always together is

The number of ways in which  $5$ men and $3$ women can sit around a round table so that every man may have at least one woman by his side, is:

$15$ persons are siting down at random at a round table, then the probability that there are $4$ persons in between two particular person $A \& B$ is

A class is composed of $2$ brothers and $8$ other boys. The combinations by which the boys can be seated at a round table so that the $2$ brothers are not seated beside each other is

The number of ways of arranging $7$ persons on a round table such that $2$ particular persons may not sit together is______.

If $N$ denotes the number of different selections of 5 letters from the word $\mathbf{W}=$ MISSISSIPPI, then N belongs to the set:

If $m$ number of integers greater than $7000$ can be formed with the digits $\mathrm{3,5},\mathrm{7,8}$ and $9,$ such that no digit is being repeated, then the value of $m$ is