Chapter 11.7, Problem 9E

To determine

To calculate: The total sets of $3$ photos.

Answer to Problem 9E

The total sets of $3$ photos possible are $4$

Explanation of Solution

Given information:

The school yearbook photographer took one photo of the school during each of the $4$ seasons and wants to choose $3$ of the photos for cover.

Calculation: To find the number of sets of $3$ photos the photographer can choose from a total of $4$ photos, use combination. As a set of $3$ photos are to be chosen from $4$ photos, the number of sets of $3$ photos can be calculated as $\text{4C3}$ .

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$

  $nCr=\frac{nPr}{r!}$   [Combination formula]

  $\Rightarrow 4C3=\frac{4P3}{3!}$   [Use combination formula]

  $=\frac{4\cdot 3\cdot 2}{3\cdot 2\cdot 1}$   [Write $4P3$ and $3!$ as a product]

  $=\frac{4\cdot \cancel{3}\cdot \cancel{2}}{\cancel{3}\cdot \cancel{2}\cdot 1}$   [Divide out common factors]

  $=4$   [Simplify]

Therefore, the number of combinations are $4$ .

To find the number of sets of $3$ photos by listing all possible combinations, name the $4$ seasons as $\text{A,B,C,D}$ . List all possible matchups. Then cross out any duplicates that represent the same matchup.

  $\begin{array}{l}\text{ABC ABD ACD}\\ \cancel{\text{BAC}}\text{ BCD }\cancel{\text{BAD}}\\ \cancel{\text{CAB}}\cancel{\text{CAD}}\cancel{\text{CBD}}\\ \cancel{\text{DAB}}\cancel{\text{DBC}}\cancel{\text{DAC}}\end{array}$

Therefore, there are $4$ possible combinations