Let `T_n` denote the number of triangles, which can be formed using the vertices of a regular polygon of `n` sides. It `T_(n+1)-T-n=21 ,t h e nn` equals a.`5` b. `7` c. `6` d. `4`

Options

A. 5
B. 7
C. 6
D. 4

Explanation

We have,
`T_(n)=""^(n)C_(3)`
`:.T_(n+1)-T_(n)=21`
`implies""^(n+1)C_(3)-""^(n)C_(3)=21`
`implies""^(n)C_(2)+""^(n)C_(3)-""^(n)C_(3)=21" "[:'""^(n)C_(r)+""^(n)C_(r-1)=""^(n+r)C_(r)]`
`implies""^(n)C_(2)=21`
`implies(n(n-1))/(2)=21`
`impliesn^(2)-n-42=0implies(n-7)(n+6)=0impliesn=7" "[:'nge1]`