Learning pathOriginal reference
Lesson 6 of 8

Prepare modular combination tables

Use factorial inverses only under their prime and range conditions.

Understand the idea

For prime p and 0 ≤ n ≤ N < p, compute C(n,r) using factorial[n] × inverseFactorial[r] × inverseFactorial[n−r] modulo p. Build factorials forward, invert the last factorial once, and fill inverse factorials backward.

Trace a small example

With p = 7, C(5,2) = 10 gives residue 3. The backward update invFact[i−1] = invFact[i] × i modulo p follows from i! = i × (i−1)!. Reuse tables only for their original modulus.

Conditions and cost

Preparation takes O(N + log p), O(N) storage, then O(1) per query. For N ≥ p, a factorial becomes zero modulo p and has no inverse. Small bounds with arbitrary positive moduli can instead use Pascal addition.

Pause & predict

Why does the simple method fail when N ≥ p?

Choose an answer
Try it yourself

Turn the idea into code

Compute C(6,2) modulo 7 and check the assumptions.

Give me a hint

6 < 7 and 7 is prime.

Show the explained solution

C(6,2) = 15, giving residue 1; the table assumptions hold.

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.