2024 ACJC P2 Q7

Taylor is planning some surprise treats for her fans during her upcoming concert. She is creating a setlist of \({12}\) songs which consists of the following:

•\({6}\) songs chosen from her entire discography which will include her number one hit song,

•\({3}\) surprise duets with a special guest artist,

•\({3}\) pre-recorded songs

1. In how many ways can Taylor arrange the setlist of \({12}\) songs by considering the following:
   • her number one hit song is the last song in the setlist,
   • the \({3}\) surprise duets are to be performed back-to-back,
   • the \({3}\) pre-recorded songs are all separated from each other by at least one song.[2]

Another segment of Taylor’s concert involves a medley consisting of dancers from different countries. It is given that her dance entourage is made up of dancers from \({6}\) different countries. There are \({5}\) dancers from each country.

2. Find the number of ways she can form a team of \({10}\) dancers from \({3}\) different countries with at least \({2}\) dancers from each country.[4]

The final segment of Taylor’s concert is a high-energy dance routine involving her most talented team of \({10}\)dancers. The choreography requires the dancers to be positioned at \({10}\) different spots on the stage. \({5}\) of the spots form a circle with each spot illuminated in blue by the spotlights. The remaining \({5}\) spots form another circle with each spot illuminated with a distinct colour by the spotlights.

3. Given that the two circles do not overlap, find the number of possible arrangements for the \({10}\) dancers at the \({10}\) spots.[2]