Learning pathOriginal reference
Lesson 2 of 8

Arrange distinct objects and ordered roles

Derive factorials and falling products from choices without replacement.

Understand the idea

Arranging n distinct objects fills positions with n, n − 1, ..., 1 choices, producing n!. Filling r ordered roles from n distinct candidates gives P(n, r) = n(n − 1)...(n − r + 1). Changing the order changes the outcome when roles differ.

Trace a small example

Choosing a captain and a deputy from five people gives 5 × 4 = 20. Selecting nobody is one empty outcome, so 0! = 1 and P(n, 0) = 1. A selection with r > n is impossible under the no-reuse rule.

Conditions and cost

Factorials grow rapidly: 20! fits in signed 64-bit, while 21! does not. Bound intermediate values, not only the final answer. Use a falling product when only r ordered choices are needed.

Pause & predict

How many ways can 4 distinct books be arranged in a row?

Choose an answer
Try it yourself

Turn the idea into code

Choose gold, silver, and bronze winners from 6 competitors with no ties.

Give me a hint

The medals are different roles.

Show the explained solution

6 × 5 × 4 = 120. Dividing by 3! would incorrectly make the ranked winners unordered.

Your notes

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Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.