2023 ACJC P2 Q8

Kelly has a box of spherical beads labelled with the \(26\) letters from A to Z, all of which have the same size. For each letter, there are exactly \(3\) beads of different colours, red, blue and gold. By stringing the labelled beads together in a specific order, Kelly plans to make a keychain. A possible design of a keychain with \(3\) labelled beads is shown in Fig. 1.

1. Find the number of ways that the labelled beads in the keychain can be arranged if it is to have
   1. \(6\) beads with distinct letters but are of the same colour.[1]
   2. \(6\) beads with \(3\) distinct letters.[3]

Kelly would like to add in beads with different shapes to the keychain. She took out another box of beads and the number of each type of beads that she has is listed in the table below:

2. Kelly chooses two beads at random from this new box of beads. The probability that both chosen beads are star-shaped but neither of those two beads are gold is \(\frac{1}{66}\).
   1. Find \(c\).[2]
   2. Find the probability that two randomly chosen beads include exactly one red bead and at least one bead with a flower shape.[3]