Chapter 12.6, Problem 46PPE

a.

To determine

To calculate: The number of line segments drawn.

Answer to Problem 46PPE

Number of line segments are $6$ .

Explanation of Solution

Given information:

  

Formula used:

A permutation is an arrangement of objects in a specific order. The expression $P{}_r$ represents the number of permutations of $n$ objects arranged $r$ at a time.

  $P{}_r=\frac{n!}{\left(n-r\right)!}$

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Number of line segments $C{}_2$ where $m$ , are the number of points.

Calculation:

According to the question,

Number of points are 4 .

Therefore,

  $\begin{array}{l}C{}_2=\frac{4!}{2!\left(4-2\right)!}\\ =\frac{4!}{2!2!}\\ =\frac{4\times 3\times 2\times 1}{2\times 1\times 2\times 1}\\ =2\times 3\times \\ =6\end{array}$

Hence, there are $6$ line segments .

b.

To determine

To calculate: The number of line segments drawn.

Answer to Problem 46PPE

Number of line segments are $45$ .

Explanation of Solution

Given information:

  

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Number of line segments $C{}_2$ where $m$ , are the number of points.

Calculation:

According to the question,

Number of points are 10 .

Therefore,

  $\begin{array}{l}C{}_2=\frac{10!}{2!\left(10-2\right)!}\\ =\frac{10!}{2!8!}\\ =\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 2\times 1}\\ =5\times 9\\ =45\end{array}$

Hence, there are $45$ line segments .

c.

To determine

To calculate: The number of points.

Answer to Problem 46PPE

Number of points are $10$ .

Explanation of Solution

Given information:

  

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Number of line segments $C{}_2$ where $m$ , are the number of points.

Calculation:

According to the question,

Number of points are 10 .

d.

To determine

To calculate: The number of segments which method to be used either permutation or combination.

Answer to Problem 46PPE

Combination is not be used.

Explanation of Solution

Given information:

  

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

Number of line segments $C{}_2$ where $m$ , are the number of points.

Calculation:

According to the question,

Number of line segments can be found by using combination . This is because we can select any point to draw a line segment.