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 $$G_{odd}(x) = \prod_{k=1}^\infty \frac{1}{1 - x^{2k-1}}.$$ This counts partitions where each part is an odd number. - The generating function for partitions into even summands is $$G_{even}(x) = \prod_{k=1}^\infty \frac{1}{1 - x^{2k}}.$$ This counts partitions where each part is an even number. 3. **Finding Coefficients:** - To find the number of partitions of 7 into odd summands, find the coefficient of $x^7$ in $G_{odd}(x)$. - To find the number of partitions of 7 into even summands, find the coefficient of $x^7$ in $G_{even}(x)$. 4. **Explicit Enumeration:** - Partitions of 7 into odd summands: - 7 - 5 + 1 + 1 - 3 + 3 + 1 - 3 + 1 + 1 + 1 + 1 - 1 + 1 + 1 + 1 + 1 + 1 + 1 Total: 5 partitions. - Partitions of 7 into even summands: - Since 7 is odd, and all summands are even, no partitions exist. Total: 0 partitions. 5. **Verification:** - The coefficient of $x^7$ in $G_{odd}(x)$ is 5, matching the explicit count. - The coefficient of $x^7$ in $G_{even}(x)$ is 0, matching the explicit count. **Final answers:** - Number of partitions of 7 into odd summands: 5 - Number of partitions of 7 into even summands: 0