a Start by finding all possible combinations in which pairs of vertices from all n vertices can be chosen.
B
b The formula for the number of combinations of n objects taken r at a time, where r ≤ n is _nC_r = n!(n-r)! * r!.
A
aExpression: _nC_2 - n
Alternative Expression: _nC_(n-2) - n
B
b n(n-3)/2
Practice makes perfect
a We want to find the expression for the number of diagonals in a convex n-sided polygon using the formula for combinations. Let's consider an example polygon with n sides.
Let's first look for the total number of ways to connect the vertices in the polygon. We want to find all possible combinations in which we can choose pairs of vertices from all n vertices. Let's recall the formula for the number of combinations of n objects taken r at a time, where r ≤ n.
_nC_r = n!/(n-r)! * r!
In our case, since we want to choose pairs r will be equal to 2. Therefore, according to the formula the total number of connections between vertices is equal to _nC_2. Notice that there are two possible types of connections.
Edges
Diagonals
In order to obtain the number of diagonals, we need to subtract the number of edges from the total number of connections between vertices. In a convex n-sided polygon there are n edges. Therefore, we are ready to write the expression for the number of diagonals in an n-sided polygon.
_nC_2 - n
Extra
Number of combinations
Notice that the numbers of combinations _nC_2 and _n C_(n-2) are the same. We will check it by comparing both expressions. Let's start by substituting n-2 for r.
Now, let's find the formula for _nC_2.
_nC_r = n!/(n-r)! * r! [0.6em]
⇓
_nC_2 = n!/(n-2)! * 2!
Both expression are the same, therefore it does not make a difference whether in the answer if we use _nC_2 and _n C_(n-2). Both formulas produce correct answers for the number of diagonals in an n-sided polygon.
b We want to use the result from Part A and find a formula for the number of diagonals. This means we need to evaluate the number of combinations _nC_2 and simplify the expression.
