Difference between revisions of "1950 AHSME Problems/Problem 45"
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==Problem== | ==Problem== | ||
| − | The number of diagonals that can be drawn in a polygon of 100 sides is: | + | The number of diagonals that can be drawn in a polygon of <math>100</math> sides is: |
<math>\textbf{(A)}\ 4850 \qquad | <math>\textbf{(A)}\ 4850 \qquad | ||
| Line 8: | Line 8: | ||
\textbf{(D)}\ 98 \qquad | \textbf{(D)}\ 98 \qquad | ||
\textbf{(E)}\ 8800</math> | \textbf{(E)}\ 8800</math> | ||
| + | |||
| + | == 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 <math>\binom{100}{2}=4950</math>. However this also counts the 100 sides of the polygon, so the actual answer is <math>4950-100=\boxed{\textbf{(A)}\ 4850 }</math>. | ||
| + | |||
| + | == Solution 2 == | ||
| + | We can choose <math>100 - 3 = 97</math> vertices for each vertex to draw the diagonal, as we cannot connect a vertex to itself or any of its two adjacent vertices. Thus, there are <math>(100)(97)/2=4850</math> diagonals, because we are overcounting by a factor of <math>2</math> (we are counting each diagonal twice - one for each endpoint). So, our answer is <math>\fbox{A}</math>. | ||
| + | |||
| + | == Solution 3== | ||
| + | The formula for the number of diagonals of a polygon with <math>n</math> sides is <math>n(n-3)/2</math>. Taking <math>n=100</math>, we see that the number of diagonals that may be drawn in this polygon is <math>100(97)/2</math> or <math>\boxed{\textbf{(A)}\ 4850 }</math>. | ||
| + | |||
| + | ~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi] | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME 50p box|year=1950|num-b=44|num-a=46}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 20:42, 3 May 2024
Problem
The number of diagonals that can be drawn in a polygon of 
sides is:
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 
. However this also counts the 100 sides of the polygon, so the actual answer is 
.
Solution 2
We can choose 
vertices for each vertex to draw the diagonal, as we cannot connect a vertex to itself or any of its two adjacent vertices. Thus, there are 
diagonals, because we are overcounting by a factor of 
(we are counting each diagonal twice - one for each endpoint). So, our answer is 
.
Solution 3
The formula for the number of diagonals of a polygon with 
sides is 
. Taking 
, we see that the number of diagonals that may be drawn in this polygon is 
or 
.
~ cxsmi
See Also
| 1950 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 44 |
Followed by Problem 46 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
