Test your understanding of binomial applications, inclusion-exclusion, pigeonhole principle, and combinatorial proofs.
1. Evaluate $^6C_0 + \\,^6C_1 + \\,^6C_2 + \\cdots + \\,^6C_6$. (1 mark)
2. In a group of 50 people, 30 like coffee, 25 like tea, and 10 like both. How many like at least one? (1 mark)
3. Show that among any 7 integers, at least two have the same remainder when divided by 6. (2 marks)
4. Prove that $^nC_r = \\,^nC_{n-r}$ algebraically. (2 marks)