QUANTITATIVE REASONING · GMAT-Q-COUNT

Counting and Discrete Arrangements

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

P (n, 2) = n (n - 1)

Translate the condition

Treat each named role as a separate choice. Apply the product rule, reducing the available choices whenever a person cannot be reused.

Test the result

Check whether order matters and whether repetition is allowed. Reversing two different role assignments creates a different outcome.

Think Like a GMAT Test-Taker

Decide whether roles or positions are distinct; that determines whether order matters.

Faster Methods

Translate the target into an equation before calculating.

Check the result against the original conditions and units.

Worked example 1

How many ordered two-person teams can be formed from 4 distinct candidates when roles are different?

P (4, 2) = 4 (3) = 12

The first named role has 4 choices and the second has 3 remaining choices, so there are 4 times 3 = 12 assignments.

Worked example 2

Four distinct reports are arranged in a row. Reports A and B must be adjacent. How many orders are possible?

3! \cdot 2! = 12

Treat the adjacent reports as one block, leaving three objects to arrange. Then account for the two internal orders of the block. A block method must count both the external arrangement and internal order.

Worked example 3

A committee of three is chosen from three analysts and four engineers. It must contain exactly one analyst. How many committees are possible?

(1 3) (2 4) = 3 \cdot 6 = 18

Choose the analyst and the two engineers separately. Since the committee has no named positions, selecting the same two engineers in a different order does not make a new committee. Multiply the independent choices.

Common Traps

Keep domain restrictions after simplifying.
Do not confuse the requested output with an intermediate quantity.