1
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Question

# Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn+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 D 7Number of triangles formed using the vertices of a regular polygon of n sides =nC3Given, Tn+1−Tn=21=>(n+1)C3−nC3=21=>nC2+nC3−nC3−21 Since, (n+1)Cr=nCr−1+nCr=>n!2!(n−2)!=2 Since (nCr=n!r!(n−r)!)=>n×(n−1)×(n−2)!2×(n−2)!=21=>n2−n=42=>n2−n−42=0=>n2−7n+6n−42=0=>n(n−7)+6(n−7)=0=>n=7 or −6Since, n cannot be negative, n=7

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