A jewellery artisan has at his disposal Brazilian stones of three colours: red, blue and green.
He intends to produce jewellery made of a metal alloy, based on a mould in the shape of a non-square rhombus with stones at its vertices, so that two consecutive vertices always have stones of different colours.
The figure illustrates a piece of jewellery, produced by this artisan, whose vertices $A$, $B$, $C$ and $D$ correspond to the positions occupied by the stones.
Based on the information provided, how many different pieces of jewellery, in this format, can the artisan obtain?
(A) 6 (B) 12 (C) 18 (D) 24 (E) 36

There are 6 students including students $\mathrm { A }$, $\mathrm { B }$, and $\mathrm { C }$. These 6 students sit around a circular table at equal intervals satisfying the following conditions. How many ways are there to seat them? (Note: arrangements that coincide by rotation are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40

There are 6 students including three students $\mathrm { A } , \mathrm { B } , \mathrm { C }$.
Find the number of ways these 6 students can all sit around a circular table with equal spacing satisfying the following conditions. (Note: rotations that coincide are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.

In how many ways can 3 couples sit around a round table such that men and women alternate and none of the couples sit together?
(a) 1
(b) 2
(c) $5! / 3$
(d) None of these.

The number of different ways to colour the vertices of a square $PQRS$ using one or more colours from the set \{Red, Blue, Green, Yellow\}, such that no two adjacent vertices have the same colour is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 84 .

Five persons $A$, $B$, $C$, $D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is\_\_\_\_

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is
(1) $6 ! 7 !$
(2) $( 6 ! ) ^ { 2 }$
(3) $( 7 ! ) ^ { 2 }$
(4) $7 !$

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
(1) 720
(2) $126 ( 5 ! ) ^ { 2 }$
(3) $7 ( 360 ) ^ { 2 }$
(4) $7 ( 720 ) ^ { 2 }$

On a circle, 12 equally spaced points are marked and numbered 1 to 12 in clockwise order. Among all triangles formed by choosing any 3 of these 12 points as vertices, the number of triangles whose three interior angles, arranged from smallest to largest, form an arithmetic sequence is (17-1)(17-2).

Six friends named Ayça, Büşra, Ceyda, Deniz, Erdem and Furkan attending a party have a table with 6 chairs around it as shown in the figure.
Ayça and Büşra, who are on bad terms, do not want to sit in chairs that are next to each other or facing each other at this table. Accordingly, in how many different ways can these six friends sit in these chairs around the table?
A) 432
B) 384
C) 360
D) 288
E) 240