2025 RI P2 Q6

Kitty has \(5\) small, \(3\) medium and \(2\) large spherical charms and each of the \(10\) charms is uniquely designed.

1. Kitty arranges all the charms in a circle on a corkboard with charms of the same sizes next to each other. How many ways are there for her to do so?[2]
2. On another occasion, Kitty arranges all the charms in a line at the base of a photo frame with none of the small charms next to each other. How many ways are there for her to do so?[2]

The diameters of all the small, medium and large charms are \(0.5\) cm, \(1\) cm and \(2\) cm respectively. Each of the spherical charms has a hole through its centre that allows it to be threaded through a chain.

3. Kitty makes a keychain that includes a \(6\) cm chain with one end attached to a keyring, as shown in Fig. 1. It is given that the \(6\) cm chain is fully threaded with charms, with no gaps between them, and is stretched taut in a straight line. For example, Fig. 2 shows the \(6\) cm chain threaded with \(2\) large and \(2\) medium charms.

How many different ways can she make a keychain with at least one charm of each of the three sizes?[4]