Permutations and Combinations 2 Question 2
2. Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is
(2019 Main, 10 April II)
(a) 180
(b) 210
(c) 170
(d) 190
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Answer:
Correct Answer: 2. (c)
Solution:
- It is given that, there are 20 pillars of the same height have been erected along the boundary of a circular stadium.
Now, the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then total number of beams $=$ number of diagonals of 20 -sided polygon.
$\because{ }^{20} C _2$ is selection of any two vertices of 20 -sided polygon which included the sides as well.
So, required number of total beams $={ }^{20} C _2-20$
$[\because$ the number of diagonals in a $n$-sided closed polygon $\left.={ }^{n} C _2-n\right ]$
$ \begin{aligned} & =\frac{20 \times 19}{2}-20 \\ & =190-20=170 \end{aligned} $
