Learning pathOriginal reference
Lesson 5 of 8

Choose groups, positions, and route steps

Translate different stories into precise combinations and independent stages.

Understand the idea

Selecting exactly k people from one group of size a and r − k from another of size b gives C(a, k)C(b, r − k). Requiring at least one from the first group is often easier as C(a + b, r) − C(b, r). For binary strings, choose the positions of the ones.

Trace a small example

A path with 3 right moves and 2 down moves is a sequence of five step positions: choose two for down, giving C(5, 2) = 10. If a mandatory grid point is reachable in the allowed directions, choose a path to it and then a path from it, multiplying the two counts.

Conditions and cost

A grid-point formula requires nonnegative step counts and unobstructed monotone movement. Check whether objects are distinct and groups are disjoint. A designated witness can overcount an “at least one” condition.

Pause & predict

How many length-5 binary strings contain exactly two 1s?

Choose an answer
Try it yourself

Turn the idea into code

Choose a 3-person team from 2 mentors and 4 students, with at least one mentor.

Give me a hint

Subtract teams made entirely of students.

Show the explained solution

C(6, 3) − C(4, 3) = 20 − 4 = 16.

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.