1. **Problem Statement:** Find the number of partitions of $n=7$ into (i) odd summands and (ii) even summands using generating functions. Then verify by listing all partitions. 2. **Generating Functions:** - The generating function for partitions into odd summands is $$G_{odd}(x) = \prod_{k=1}^\infty \frac{1}{1 - x^{2k-1}}$$ - The generating function for partitions into even summands is $$G_{even}(x) = \prod_{k=1}^\infty \frac{1}{1 - x^{2k}}$$ 3. **Key Rule:** The coefficient of $x^n$ in these generating functions gives the number of partitions of $n$ into the specified summands. 4. **Partitions of 7 into odd summands:** - List all partitions of 7 with odd parts: - 7 - 5 + 1 + 1 - 3 + 3 + 1 - 3 + 1 + 1 + 1 + 1 - 1 + 1 + 1 + 1 + 1 + 1 + 1 - Count: 5 partitions. 5. **Partitions of 7 into even summands:** - Since 7 is odd, partitions into only even summands are impossible (sum of even numbers is always even). - Count: 0 partitions. 6. **Verification via generating functions:** - Coefficient of $x^7$ in $G_{odd}(x)$ is 5. - Coefficient of $x^7$ in $G_{even}(x)$ is 0. **Final answers:** - Number of partitions of 7 into odd summands: **5** - Number of partitions of 7 into even summands: **0**