Permutations and Combination

A Diagonal Of a polygon is defined to be a line joining any two non-adjacent vertices.
a) Show that the number of diagonals in a 5-sided polygon
is (5 2) - 5
b) how many diagonals are there in 6-sided polygon?
c) show that the number of diagonals in an n-sided polygon is n(n-3) /2

Original post by Sameer321

A Diagonal Of a polygon is defined to be a line joining any two non-adjacent vertices.
a) Show that the number of diagonals in a 5-sided polygon
is (5 2) - 5
b) how many diagonals are there in 6-sided polygon?
c) show that the number of diagonals in an n-sided polygon is n(n-3) /2

What have you tried?

For part (a), notice that 5C2 is just the number of ways to choose any two vertices out of five. But what this includes are also choices where vertices are adjacent, we don't want those. Can you see how there are exactly 5 choices we needs to exclude where the vertices are adjacent?

(edited 4 years ago)

Reply 2

Sameer321

OP

6

So for Part b
It should be 6C2 - 6?
And please give hint of part 3?

Original post by Sameer321

So for Part b
It should be 6C2 - 6?
And please give hint of part 3?

Yep. Can you see what the pattern is?? Can you follow this pattern through for an n-sided polygon?

For 5-sided polygon, you did

\displaystyle \binom{5}{2} - 5

For 6-sided polygon, you did

\displaystyle \binom{6}{2} - 6

...

For n-sided polygon, you do (?????)

If you convert this (????) form into factorial form, things cancel and you get left with an expression you can simplify into

\dfrac{n(n-3)}{2}

.

(edited 4 years ago)

Permutations and Combination

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