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# The formula 180(n-2) gives the number of degrees

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The formula 180(n-2) gives the number of degrees in the angles of a convex polygon because n-2 triangles can be drawn (with no lines crossing) in a polygon with n sides, each triangle containing 180. In how many ways can a convex heptagon be divided into five triangles if each divided into five triangles if each different orientation is counted separately? (Hint: look up the Catalan sequence)

a) 6

b) 36

c) 42

d) 48

e) 52

Guest Jan 21, 2018
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The formula 180(n-2) gives the number of degrees in the angles of a convex polygon because n-2 triangles can be drawn (with no lines crossing) in a polygon with n sides, each triangle containing 180. In how many ways can a convex heptagon be divided into five triangles if each divided into five triangles if each different orientation is counted separately? (Hint: look up the Catalan sequence)

a) 6

b) 36

c) 42

d) 48

e) 52

Convex heptagon:

$$C_5 \text{ is the number of different ways a convex polygon with } 5 + 2 =7 \text{ sides can be cut into triangles }.$$

The nth Catalan number is given directly in terms of binomial coefficients by

$$\begin{array}{|rcll|} \hline C_n = \dfrac{1}{n+1}\dbinom{2n}{n} \\ \hline \end{array}$$

$$\begin{array}{rcll} \mathbf{C_5} &=& \dfrac{1}{5+1}\dbinom{2\cdot 5}{5} \\ &=& \dfrac{1}{6}\dbinom{10}{5} \\ &=& \dfrac{1}{6}\cdot \dfrac{10}{5} \cdot \dfrac{9}{4}\cdot \dfrac{8}{3}\cdot \dfrac{7}{2}\cdot \dfrac{6}{1} \\ &=& 2\cdot 3\cdot 7 \\ &\mathbf{=}& \mathbf{42} \\ \end{array}$$

A convex heptagon can be divided into five triangles in 42 ways  (with no lines crossing).

heureka  Jan 22, 2018
edited by heureka  Jan 22, 2018