Ten points are marked on a circle. The number | vedclass

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(C) polygon is formed by selecting $n$ points out of $10$ points,where $n \ge 3$.The number of ways to select $n$ points from $10$ is given by the com

Vedclass · Accepted Answer

$968$

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(C) polygon is formed by selecting $n$ points out of $10$ points,where $n \ge 3$.The number of ways to select $n$ points from $10$ is given by the combination formula $\binom{10}{n}$.Since any selection of $n$ points $(n \ge 3)$ on a circle forms exactly one unique convex polygon,the total number of such polygons is the sum of combinations for $n = 3, 4, \dots, 10$.Total polygons = $\binom{10}{3} + \binom{10}{4} + \binom{10}{5} + \binom{10}{6} + \binom{10}{7} + \binom{10}{8} + \binom{10}{9} + \binom{10}{10}$.We know that $\sum_{n=0}^{10} \binom{10}{n} = 2^{10} = 1024$.Therefore,$\sum_{n=3}^{10} \binom{10}{n} = 2^{10} - \binom{10}{0} - \binom{10}{1} - \binom{10}{2}$.$= 1024 - 1 - 10 - \frac{10 	imes 9}{2} = 1024 - 1 - 10 - 45 = 1024 - 56 = 968$.

Ten points are marked on a circle. The number of distinct convex polygons of three or more sides that can be drawn using some or all of the ten points as vertices is:

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