First slide

Combinations

Question

Let $T_{n}$ denote the number of triangles which can be formed by using the vertices of a regular polygon of $n$ sides. If $T_{n + 1} - T_{n} = 21$ , then $n$ equals

Moderate

A

5
B

7
C

6
D

4

Solution

The number of triangles that can be formed by using the vertices of a regular polygon is $^{n} C_{3}$ . That is $T_{n} =^{n} C_{3}$
Now, $T_{n + 1} - T_{n} = 21$
$\Rightarrow^{n + 1} C_{3} -^{n} C_{3} = 21$
$\Rightarrow^{n} C_{2} +^{n} C_{3} -^{n} C_{3} = 21 [∵^{n + 1} C_{r} =^{n} C_{r - 1} +^{n} C_{r}]$
$\Rightarrow \frac{1}{2} n (n - 1) = 21 \Rightarrow n = - 6 or 7$
As $n$ is a positive integer, $n$ = 7.

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