Question

How many pentagons can be made by vertices of dodecagon, such that no side of pentagon coincide with
side of dodecagon ?
(a) 36
(b) 56
O (c) 46
(d) 26​

Answer 1

no. of ways to select r items out of n consecutive items such that no two consecutive itens selected = n-r+1 C r

Answer 2

Concept Introduction:-

Upon closer inspection, the pentagon appeared to be a five-sided geometric shape.

Given Information:-

We have been given that a question.

To Find:-

We have to find that number of pentagons can be made by vertices of dodecagon, such that no side of pentagon coincide with side of dodecagon.

Solution:-

According to the problem

No. of ways to select $5$ vertices (for Pentagon) $12$ Consecutive vortices-such that no two ve of dodecagon are consecutive $=12-5+1_{c}_{_5} }$

$={ }^{8} c_{5}$

No. in $3$ to $10=10-3+1=8$

No. of ways to select $3$ vertices out of $8$ consecutive vertices. Such that no two vertices are (o

$=8-3+1 c_{3}=& 6_c__{3}$

$={ }^{8} c_{5}-{ }^{6} c_{3}}$

$\frac{8*7*6}{3*2*1}- \frac{6*5*4}{3*2*1}\\=56-20\\=36$.

Final Answer:-

The right option is (a) $36$.