Kitty has $5$ small, $3$ medium and $2$ large spherical charms and each of the $10$ charms i

Question

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

Accepted Answer

(a)2880 (b)86400 (c)169200

2025 RI P2 Q6

2025 RI P2 Q6