Decision and conditions
Count unordered selections, repeated objects and forbidden arrangements separately. People are distinct; selection order is irrelevant unless roles are assigned. Repeated symbols are indistinguishable.
Worked model
A committee of three is chosen from 7 distinct people. How many committees are possible? The ordered selection count is 7(6)(5). Each committee occurs in 3! = 6 orders. Divide by six to obtain 35. Before calculating: Different orders of selection can describe the same unordered committee.
Contrast: a different decision
A committee of three is chosen from 7 people and must contain a designated member. How many committees are possible? The designated member occupies one of the three seats. Choose two from the remaining people without ordering them. The count is (6)(5)/2 = 15. Before calculating: Reserve the required member before selecting the remaining seats.
Transfer with solution
How many distinct strings can be formed by rearranging 3 identical A symbols and two identical B symbols? There are 5 positions total. Choose exactly two positions for the Bs. The count is (5)(4)/2 = 10. Before calculating: Choose the B positions; exchanges of identical As or Bs do not create new strings.
Check the boundary
Subtract only genuinely forbidden cases from an unrestricted count. 5 distinct books, including A and B, are arranged in a row. How many arrangements have A and B not adjacent? All arrangements: 5! = 120. Adjacent arrangements: 2(4)! = 48. The permitted count is 72.
A committee of three is chosen from 7 distinct people. How many committees are possible?
- The ordered selection count is 7(6)(5).
- Each committee occurs in 3! = 6 orders.
- Divide by six to obtain 35.
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.
GRE Quantitative Reasoning overview