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Lesson 8 of 8

Handle repeated objects and arithmetic safely

Extend the model to repeated letters, circles, distributions, and exact calculations.

Understand the idea

Repeated indistinguishable letters divide a distinct arrangement count by each frequency factorial. Circular arrangements of distinct objects give (n − 1)! only when rotations are identified and reflections are still different. Stars and bars counts S identical objects in K labelled boxes as C(S + K − 1, K − 1) for K ≥ 1 and nonnegative contents.

Trace a small example

AAB has 3!/2! = 3 arrangements. Distributing 3 identical objects into 2 labelled boxes has 4 solutions: (0,3), (1,2), (2,1), (3,0). Lower bounds can be removed by assigning them first; upper bounds require excluding invalid assignments rather than ignoring them.

Conditions and cost

Exact integer division and division modulo M are different operations. Promote before multiplication and check intermediate bounds. For large exact combinations use proved cancellation; for small arbitrary-modulus counts Pascal addition is safer than dividing residues.

Pause & predict

How many distinct rearrangements of AAB exist?

Choose an answer
Try it yourself

Turn the idea into code

Distribute 5 identical objects among 3 labelled boxes, with at least one in each.

Give me a hint

Put one in each box first, then distribute the remaining two.

Show the explained solution

The remaining nonnegative allocations number C(2 + 3 − 1, 3 − 1) = C(4, 2) = 6.

Your notes

Ready for the next idea?

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.