1950 AHSME Problems/Problem 45

Problem

The number of diagonals that can be drawn in a polygon of 100 sides is:

$\textbf{(A)}\ 4850 \qquad \textbf{(B)}\ 4950\qquad \textbf{(C)}\ 9900 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 8800$

Solution

Each diagonal has its two endpoints as vertices of the 100-gon. Each pair of vertices determines exactly one diagonal. Therefore the answer should be $\binom{100}{2}=4950$. However this also counts the 100 sides of the polygon, so the actual answer is $4950-100=\boxed{\textbf{(A)}\ 4850 }$.

Solution 2

For each vertex we can choose $100 - 3 = 97$ vertices to draw the diagonal, as we cannot connect a vertex to itself or either adjacent vertices. Thus, the answer is $(100)(97)/2=4850$, as we are overcounting by a factor of $2$ (we are counting each diagonal twice - one for each endpoint). Thus our answer is $\fbox{A}$.

See Also

1950 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 44
Followed by
Problem 46
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