1. **State the problem:** We need to show two binomial coefficient identities: a) $\binom{n}{1} = n$ b) $\binom{n}{2} = \frac{n(n-1)}{2}$ 2. **Recall the definition of binomial coefficients:** The binomial coefficient $\binom{n}{k}$ is defined as: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ 3. **Prove part (a):** Substitute $k=1$: $$\binom{n}{1} = \frac{n!}{1!(n-1)!} = \frac{n \times (n-1)!}{1 \times (n-1)!} = n$$ So, $\binom{n}{1} = n$ as required. 4. **Prove part (b):** Substitute $k=2$: $$\binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n \times (n-1) \times (n-2)!}{2 \times 1 \times (n-2)!} = \frac{n(n-1)}{2}$$ Thus, $\binom{n}{2} = \frac{n(n-1)}{2}$ as required. 5. **Summary:** We used the factorial definition of binomial coefficients and simplified to prove both identities. **Final answers:** $$\binom{n}{1} = n$$ $$\binom{n}{2} = \frac{n(n-1)}{2}$$