$3.$ Given the dataset in the excel file, you want to apply a K$-$means clustering method to cluster the data, but first you need to use Silhouette method to find out how many clusters are there. $(40$ points$)$

a$)$ Write out the formula and explain what is Silhouette index $(10$ pt$)$

a$)$ selection of $\backslash ($ K $\backslash )$ based on Silhouette index values for $\backslash ($ K$=2,3,4,5\backslash )(20$ pt $)$

Provide codes and step by step output plots or prints $($e$.$g$,$ silhouette plot, Silhouette values$)$

c$)$ Apply selected K$-$means to the data for clustering $(10$ pt $)$

Provide codes and output plots $($clustering output$)$

Example Silhouette plot: