Circular Permutation

Q. The number of ways can 7 boys be seated at a round table so that 2 particular boys are next to each other is:-

1. $6!2!$
2. $6!$
3. $5!2!$
4. $5!$

Q. The number of ways in which 8 red roses and 5 white roses of different sizes can be made out to form a garland so that no two white roses come together is

1. 
2. 
3. 
4. 

Q. The posted weight limit for a covered wooden bridge in Pennsylvania is $6000$ pounds. A delivery truck that is carrying $x$ identical boxes each weighing $14$ pounds will pass over the bridge. If the combined weight of the empty delivery truck and its driver is $4500$ pounds, what is the maximum possible value for $x$ that will keep the combined weight of the truck, driver, and boxes below the bridge's posted weight limit?

1. $107$
2. $108$
3. $105$
4. $106$

Q. In Figure 1, the length of $\overline{PS}$ is $2x+12$, and the length of $\overline{PQ}$ is $6x-10$. If $R$ is the midpoint of $\overline{QS}$, calculate the length of $\overline{PR}$.

1. $-4+2x$
2. $-2x+11$
3. $4x+1$
4. $2x+2$

Q. Degree of unsaturation for is?

1. $9$
2. $6$
3. $8$
4. $7$

Q. Find the number of ways of arranging $6$ boys and $6$ girls arround a circular table so that

1. all the girls sit together
   
2. no two girls sit together
   
3. boys and girls sit alternately