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r-sided polygons are formed by joining t...

`r`-sided polygons are formed by joining the vertices of an `n`-sided polygon. Find the number of polygons that can be formed, none of whose sides coincide with those of the `n`-sided polygon.

Text Solution

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`y_1-y_2>=0`
`y_2-y_3.....y_(r-1)>=1`
`y_1+y_2+y_3....y_r-1>=1`
`y_1+y_2+y_3.....y_r=N-2r`
`=N-2r+r-1_(C_(r-1))=N-r-1_(C_(r-1))`
Total polygons=`N-r-1_(C_(r-1))+N-r_(C_(r))`.
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