c$.$ A dataset $S$ was collected and presented in the figure below. Each data sample has two features $x1$ and $x2$ and is classified as either cross or circle. You are asked to use Binary Decision Trees to build a binary classification model. $(1)$ Based on the calculation of entropy$-$based information gain, at the root condition node, is it better to use $x1$ with the threshold of $10$ or $2$ with the threshold of $15?(2)$ Provide detailed calculations to support your answer. The formulas of Entropy and Information Gain are provided below.

Entropy: $,H(S)={\displaystyle \underset{i=1}{\overset{C}{}}}-P_{i}lo{g}_2(P_i),$ dataset $S$ contains $C$ classes.

Information Gain: $IG(S,x)=H(S)-{\displaystyle \underset{\text{t}\text{i}\text{n}\text{T}}{\overset{?}{}}}P(t)H(t),$ where $S=U{t}_{\text{t}\text{i}\text{n}\text{T}}$