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I am not entirely sure what your question is, but here is my interpretation of it: is every $n$-manifold $X$ (without boundary) the boundary of some $(n+1)$-manifold with boundary $Y$? The answer is yes: just take $Y=X\times [0,\infty)$, identifying $X$ with $\partial Y=X\times\{0\}$. (Mike Miller gave an answer to the contrary in the comments; however, ...

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It is a consequence of the co-area formula. In particular, I studied it in the books by Giaquinta and Modica, volumes 4 and 5.

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For combinatorics, I would highly recommend the book by Yao Zhang. Its an excellent wealth of information and provides common strategies for counting that tackle pretty much all problems out there. It has both text that teaches you, example problems, and "challenge" problems. You can find it easily online.

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