Chapter 12.9, Problem 26E

a.

To determine

Find the formula to calculate the total number of interactions that should take place if $n$ people participate.

Answer to Problem 26E

  $X_n=\frac{1}{2}\left(n-1\right)n$

Explanation of Solution

Given information:

Mellissa needs to coordinate a “get to know you” activity for the group’s first meeting. The activity must last at least $40\mathrm{minutes}$ but no more than $60\mathrm{minutes}$ . Mellissa had chosen an activity that will require each participant to interact with every other participant in attendance only once and estimates that if her directions are followed, each interaction should take approximately $30\mathrm{seconds}$ .

Calculation:

Here, we will sample some groups of $2,3,4$ people count the interaction arrive at a hypothesis.

  $\therefore 1person=0\mathrm{interactions}$

Now, we will count the interactions $X$ by naming people with $1,2,3etc$ .

Thus, interaction $34$ is the same as $43$ as it is the interaction between $person3andperson4$ .

  $\begin{array}{l}2people:\\ Interactions=12\\ 2people:\\ Interactions=12\\ Total=1\\ n=2,X=1\\ 3people\left(1,2and3\right):\\ Total=2+1=3\\ n=3,X=3\\ 4people:\\ Interactions=12,13,14,23,24,34\\ Total=3+2+1=6\\ n=4,X=6\\ 5people:\\ Interactions=12,13,14,15,23,24,25,34,35,45\\ Total=4+3+2+1=10\\ n=5\\ 1+2+3+4=\frac{1}{2}4\left(4+1\right)\\ Thesumofthefirst\left(n-1\right)\mathrm{integers}:\\ X_n=\frac{1}{2}\left(n-1\right)n\end{array}$

Hence, the required answer is $X_n=\frac{1}{2}\left(n-1\right)n$ .

b.

To determine

Show that the formula is valid for all positive integral values of $n$ .

Answer to Problem 26E

The formula is valid.

Explanation of Solution

Given information:

Mellissa needs to coordinate a “get to know you” activity for the group’s first meeting. The activity must last at least $40\mathrm{minutes}$ but no more than $60\mathrm{minutes}$ . Mellissa had chosen an activity that will require each participant to interact with every other participant in attendance only once and estimates that if her directions are followed, each interaction should take approximately $30\mathrm{seconds}$ .

Calculation:

Here, we will use mathematical induction to prove the given situation.

Now, when $1person$ is in the group, there is no one to interact with so $X_1=0$

Thus, the formula gives $\frac{1}{2}\left(1-1\right)=0$ so the statement is valid for $n=1$

Now, suppose that the statement is true for $n=k$ as when there is $k$ people in the group that the number of interactions is:

  $X_k=\frac{1}{2}\left(k-1\right)k$

  $\begin{array}{l}{X}_k=\frac{1}{2}\left(k-1\right)k\\ X_{k+1}=X_k+k\\ X_{k+1}=\frac{1}{2}\left(k-1\right)k+k\\ X_{k+1}=\frac{k\left(k-1\right)+2k}{2}\\ X_{k+1}=\frac{k^2-k+2k}{2}\\ X_{k+1}=\frac{k^2+k}{2}\\ X_{k+1}=\frac{1}{2}k\left(k+1\right)\\ Thestatementisvalidforn=k+1\end{array}$

Hence, the formula is valid.

c.

To determine

If $15people$ participate then will Mellissa’s activity meet the guidelines provided to her.

Answer to Problem 26E

It will meet the guidelines.

Explanation of Solution

Given information:

Mellissa needs to coordinate a “get to know you” activity for the group’s first meeting. The activity must last at least $40\mathrm{minutes}$ but no more than $60\mathrm{minutes}$ . Mellissa had chosen an activity that will require each participant to interact with every other participant in attendance only once and estimates that if her directions are followed, each interaction should take approximately $30\mathrm{seconds}$ .

Calculation:

Here,

  $\begin{array}{l}{X}_{15}=\frac{1}{2}\left(14\right)\left(15\right)=105\\ Thetotaltimeneededis:\\ \frac{105}{2}=50.5\mathrm{minutes}\end{array}$

Thus, it is within the project constraint of $40-60\mathrm{minutes}$ .

Hence, activity will meet the guidelines.