Can anyone please walk me through how to solve problems like this?
1) how many diagonals does an n-gon have?
2) can a kindgom in which 77 roads lead out of each city, have exactly 10000 roads?
Can anyone please walk me through how to solve problems like this?
1) how many diagonals does an n-gon have?
2) can a kindgom in which 77 roads lead out of each city, have exactly 10000 roads?
(1) There is a bijection between diagonals and pairs of non-adjacent vertices of the $n$-gon. There are $\binom{n}2$ pairs of vertices altogether, of which $n$ pairs consist of adjacent vertices, so there are $\binom{n}2-n$ pairs of non-adjacent vertices the same number of diagonals. If you want, you can expand this to get rid of the binomial coefficient:
$\binom{n}2-n=\frac{n(n-1)}2-n=\frac{n^2-n}2-n=\frac{n^2-3n}2=\frac{n(n-3)}2$
(and you can pick your favorite form).
(2) In graph-theoretic terms this is asking whether a graph in which every vertex has degree $77$ can have exactly $10000$ edges. Each edge has two ends, so the sum of the degrees of the vertices in any graph is twice the number of edges. If our kingdom has $n$ cities (and our graph therefore $n$ vertices), the sum of the degrees will be $77n$. If there were $10000$ roads (edges), we’d have $77n=2\cdot10000$; is this actually possible?