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Question

# Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon on n sides. If Tnâˆ’1âˆ’Tn=21, then n equals

A
5
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B
7
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C
6
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D
4
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Solution

## The correct option is B 7Number of triangles that can be formed by using the vertices of a regular polygon=nc3Now Tn+1−Tn=21n+1c3−nc3=21⇒nc2+nc3−nc3=21 since n+1cr=ncr−1+ncr⇒12n(n−1)=21⇒n2−n=42⇒n2−n−42=0⇒n2−7n+6n−42=0⇒n(n−7)+6(n−7)=0⇒n+6=0,n−7=0∴n=−6,7As n is a positive integer, we have n=7

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