Question
XYZ Corporation hired 6 people as summer workers. Of these 6 people, 3 will be hired as permanent employees in the fall. If 2 of the 6 people are male, how many of the possible groups of 3 people hired in the fall will include only 1 male?
- A4
- B6
- C12
- D24
Show the answer
Correct answer: C
An organized list can be used to determine the number of groups of 3 people that include only 1 male.
Let the letters A, B, C, and D represent the females, and let the numbers 1 and 2 represent the males.
Then, create a list of all possible combinations of 3 people with only 1 male.
AB1, AB2, AC1, AC2, AD1, AD2
BC1, BC2, BD1, BD2
CD1, CD2
There are 12 possible groups with only 1 male.
Question analysis
A company hired 6 summer workers, 2 of them male, and will keep 3 as permanent employees in the fall. This counting question asks how many possible groups of 3 include only 1 male. Students count combinations where order does not matter, and the bold word only, with two kinds of workers to combine, makes careful listing essential.
- NYS standards
- 7.SP.8b sample spaces for compound events
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 students list outcomes with organized lists and tree diagrams, usually ordered ones. Counting unordered groups of three under an "only one" condition, built from two kinds of people, goes beyond that typical work.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Combines ideas moderate
- Something you must already know minor
- Turning words into math minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 7, 41 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
The question hides a two-stage count in a hiring story. Students tend to get one stage right and forget the other: they count the pairs of women and stop, or they count ordered picks and so find every group twice. Naming the women A to D and the men 1 and 2, then listing the pairs of women alphabetically with each man attached, makes every group visible and checkable.
Combines ideas moderate Working it out
Two counts are joined: the 6 ways to pick 2 of the 4 women, then the 2 ways to pick the man, and the total is their product. (Embretson, 1998; Daniel & Embretson, 2010)
Something you must already know minor Working it out
Students must already know that a group is the same whichever order its members are named, so picking one woman and then another repeats the reverse pick. (Anderson & Krathwohl, 2001)
Turning words into math minor Understanding the question
The bold word only means exactly one man, so the other two places in each group of 3 must go to two of the 4 women, a step the stem leaves unsaid. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Strategy: Name the women A to D and the men 1 and 2, list every pair of women without repeats, then attach each man to each pair and count the groups.
Traps
- Stops one step early: A choice is a number from partway through the solution.
- Overlapping groups: Things in two groups at once are counted twice, or not subtracted.
The wrong answers, classified
A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Multiplies the 2 men by the 2 places left for women, 2 × 2 = 4, using numbers from the question without listing a single group.
B Stops one step early (King et al., 2004; Gierl et al., 2017)
Counts the 6 pairs of women correctly but stops before pairing each one with either of the 2 men, which doubles the count.
D Overlapping groups (King et al., 2004)
Counts ordered picks of the two women, 4 × 3 = 12 instead of 6, so each group appears twice, then doubles for the men to reach 24.
Worked solution, step by step
Formulas used
- Counting with two choices
- Each way of picking the man can go with each way of picking the two women. So you multiply the two counts. It is like outfits: 2 shirts with 6 pairs of pants make 12 outfits.
Sort out the question
Six people work for the summer. Two are male, so the other 4 are female. Three will be hired in the fall.
You need the number of groups of 3 with only 1 male. That means exactly 1 man and 2 women. In a group, order does not matter. Ann, Bea and Carl is the same group as Carl, Ann and Bea.
Give each person a name
Call the women A, B, C and D. Call the men 1 and 2. Every group you want has 2 letters and 1 number, like AB1.
List the pairs of women
List each pair once. Start with A, then pair B only with letters after it, and so on:
- AB, AC, AD
- BC, BD
- CD
That is 6 pairs. BA is not new, since it is the same pair as AB.
Watch out: counting the order gives pairs, each one twice. Then doubling for the men gives 24, choice D.
Add a man to each pair
Each pair of women can go with man 1 or man 2. Use the counting rule:
Watch out: stopping at the 6 pairs of women leaves out the men. That gives 6, choice B.
So the answer is C, 12.
Check your answer
Write out every group. With man 1: AB1, AC1, AD1, BC1, BD1, CD1. With man 2: AB2, AC2, AD2, BC2, BD2, CD2. Each list has 6 groups, and . No group repeats, and each has just one man.
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