Decision and conditions
Reserve required members and subtract forbidden cases without ordering a committee. People are distinct; repeated symbols are not. Unordered selections count each group once.
Worked example
A three-person committee is chosen from 7 distinct people, including Alex and Blair. Exactly one of Alex and Blair must be on the committee. How many committees are possible? There are two choices for the single designated member. Choose two more from the other 5 people: C(5,2)=10. The number is 2×10=20. Method choice: Choose which designated person is included, exclude the other, then choose the remaining seats.
Contrast: change the reasoning
How many distinct strings contain exactly 3 identical As and two identical Bs if the first and last symbols must be different? Both A…B and B…A are allowed. The endpoint orders are A…B and B…A. For either order, the interior has 3 positions containing one B and 2 identical As. Choose the B position in each order: 2×3=6. Method choice: Choose the two different endpoint symbols, then place the remaining B among the interior positions.
Transfer with solution
5 distinct books, including A and B, are arranged in a row. Exactly one book must lie between A and B; either order A…B or B…A is allowed. How many arrangements are possible? The left position of the pair can be 1 through 3. A and B can occupy those positions in two orders. The other 3 distinct books fill the remaining positions in 6 ways; total 36. Method choice: Choose positions two places apart for A and B, choose their order, then fill all remaining positions.
Boundary check
A two-person team is chosen from 3 women and three men. How many teams include at least one woman? All unordered pairs: (6)(5)/2. There are three all-male pairs. Subtract them to obtain 12. Method choice: The complement consists of teams made entirely of men.
Misconception to reject
Choosing at least one member of a category includes multiple category counts. Ignores the restriction on the two designated people.
A three-person committee is chosen from 7 distinct people, including Alex and Blair. Exactly one of Alex and Blair must be on the committee. How many committees are possible?
- There are two choices for the single designated member.
- Choose two more from the other 5 people: C(5,2)=10.
- The number is 2×10=20.
Compare the other answer choices
Counts only committees with both designated people, the excluded case.
Includes only Alex and omits the symmetric committees containing only Blair.
Chooses a designated person and then allows the other to occupy a remaining seat, also counting both-designated committees twice.
Ignores the restriction on the two designated people.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam content