一、Prove by mathematical induction that \[ (1+x)^n = C_0^n + C_1^n \times x + C_2^n \times x^2 + \dots + C_n^n \times x^n, n \in \mathbb{N}. \](20%)

四、Paul has two coolers. Cooler 1 contains 8 cans of cola and 3 cans of lemonade. Cooler 2 contains 5 cans of cola and 7 cans of lemonade. Paul randomly selects one can from Cooler 1 and transfers it into Cooler 2. Subsequently, Betty randomly selects two cans from Cooler 2. Given that both of Betty's selections are cans of cola, what is the probability that the can Paul initially transferred was lemonade? (10%)

一、Prove by mathematical induction that

\[ (1+x)^n = C_0^n + C_1^n \times x + C_2^n \times x^2 + \dots + C_n^n \times x^n, n \in \mathbb{N}. \]
(20%)

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