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If $T_n$ denotes the number of triangles which can be formed using the vertices of regular polygon of $\mathrm{n}$ sides and $\mathrm{T}_{\mathrm{n}+1}-\mathrm{T}_{\mathrm{n}}=21$, then $\mathrm{n}=$
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Verified Answer
The correct answer is:
7
According to the given condition, $\mathrm{T}_{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_3$
$$
\therefore \quad \mathrm{T}_{\mathrm{n}+1}-\mathrm{T}_{\mathrm{n}}=21 \Rightarrow{ }^{\mathrm{n}+1} \mathrm{C}_3-{ }^{\mathrm{n}} \mathrm{C}_3=21
$$
Note that $\mathrm{n}=7$ satisfies the above condition.
$\therefore \quad$ Option (B) is correct.
$$
\therefore \quad \mathrm{T}_{\mathrm{n}+1}-\mathrm{T}_{\mathrm{n}}=21 \Rightarrow{ }^{\mathrm{n}+1} \mathrm{C}_3-{ }^{\mathrm{n}} \mathrm{C}_3=21
$$
Note that $\mathrm{n}=7$ satisfies the above condition.
$\therefore \quad$ Option (B) is correct.
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