C++ – How to we compute N choose K modulus a prime number without overflow

I see you're using unsigned integers. By definition, in C (I don't know about C++), unsigned arithmetic does not overflow ... so, at least for C, your point is moot :)

With signed integers, once there has been overflow, undefined behaviour (UB) has occurred and your program can do anything (for example: render tests inconclusive). 

#include <limits.h>

int a = <something>;
int x = <something>;
a += x;              /* UB */
if (a < 0) {         /* Unreliable test */
  /* ... */
}

To create a conforming program, you need to test for overflow before generating said overflow. The method can be used with unsigned integers too:

// For addition
#include <limits.h>

int a = <something>;
int x = <something>;
if ((x > 0) && (a > INT_MAX - x)) /* `a + x` would overflow */;
if ((x < 0) && (a < INT_MIN - x)) /* `a + x` would underflow */;

// For subtraction
#include <limits.h>
int a = <something>;
int x = <something>;
if ((x < 0) && (a > INT_MAX + x)) /* `a - x` would overflow */;
if ((x > 0) && (a < INT_MIN + x)) /* `a - x` would underflow */;

// For multiplication
#include <limits.h>

int a = <something>;
int x = <something>;
// There may be a need to check for -1 for two's complement machines.
// If one number is -1 and another is INT_MIN, multiplying them we get abs(INT_MIN) which is 1 higher than INT_MAX
if ((a == -1) && (x == INT_MIN)) /* `a * x` can overflow */
if ((x == -1) && (a == INT_MIN)) /* `a * x` (or `a / x`) can overflow */
// general case
if (a > INT_MAX / x) /* `a * x` would overflow */;
if ((a < INT_MIN / x)) /* `a * x` would underflow */;

For division (except for the INT_MIN and -1 special case), there isn't any possibility of going over INT_MIN or INT_MAX.

Beginner

Introductory, no previous programming experience

• C++ Primer * (Stanley Lippman, Josée Lajoie, and Barbara E. Moo) (updated for C++11) Coming at 1k pages, this is a very thorough introduction into C++ that covers just about everything in the language in a very accessible format and in great detail. The fifth edition (released August 16, 2012) covers C++11. [Review]

  ^{* Not to be confused with C++ Primer Plus (Stephen Prata), with a significantly less favorable review.}

• Programming: Principles and Practice Using C++ (Bjarne Stroustrup, 2nd Edition - May 25, 2014) (updated for C++11/C++14) An introduction to programming using C++ by the creator of the language. A good read, that assumes no previous programming experience, but is not only for beginners.

Introductory, with previous programming experience

• A Tour of C++ (Bjarne Stroustrup) (2nd edition for C++17) The “tour” is a quick (about 180 pages and 14 chapters) tutorial overview of all of standard C++ (language and standard library, and using C++11) at a moderately high level for people who already know C++ or at least are experienced programmers. This book is an extended version of the material that constitutes Chapters 2-5 of The C++ Programming Language, 4th edition.

• Accelerated C++ (Andrew Koenig and Barbara Moo, 1st Edition - August 24, 2000) This basically covers the same ground as the C++ Primer, but does so in a quarter of its space. This is largely because it does not attempt to be an introduction to programming, but an introduction to C++ for people who've previously programmed in some other language. It has a steeper learning curve, but, for those who can cope with this, it is a very compact introduction to the language. (Historically, it broke new ground by being the first beginner's book to use a modern approach to teaching the language.) Despite this, the C++ it teaches is purely C++98. [Review]

Best practices

• Effective C++ (Scott Meyers, 3rd Edition - May 22, 2005) This was written with the aim of being the best second book C++ programmers should read, and it succeeded. Earlier editions were aimed at programmers coming from C, the third edition changes this and targets programmers coming from languages like Java. It presents ~50 easy-to-remember rules of thumb along with their rationale in a very accessible (and enjoyable) style. For C++11 and C++14 the examples and a few issues are outdated and Effective Modern C++ should be preferred. [Review]

• Effective Modern C++ (Scott Meyers) This is basically the new version of Effective C++, aimed at C++ programmers making the transition from C++03 to C++11 and C++14.

• Effective STL (Scott Meyers) This aims to do the same to the part of the standard library coming from the STL what Effective C++ did to the language as a whole: It presents rules of thumb along with their rationale. [Review]

Intermediate

• More Effective C++ (Scott Meyers) Even more rules of thumb than Effective C++. Not as important as the ones in the first book, but still good to know.

• Exceptional C++ (Herb Sutter) Presented as a set of puzzles, this has one of the best and thorough discussions of the proper resource management and exception safety in C++ through Resource Acquisition is Initialization (RAII) in addition to in-depth coverage of a variety of other topics including the pimpl idiom, name lookup, good class design, and the C++ memory model. [Review]

• More Exceptional C++ (Herb Sutter) Covers additional exception safety topics not covered in Exceptional C++, in addition to discussion of effective object-oriented programming in C++ and correct use of the STL. [Review]

• Exceptional C++ Style (Herb Sutter) Discusses generic programming, optimization, and resource management; this book also has an excellent exposition of how to write modular code in C++ by using non-member functions and the single responsibility principle. [Review]

• C++ Coding Standards (Herb Sutter and Andrei Alexandrescu) “Coding standards” here doesn't mean “how many spaces should I indent my code?” This book contains 101 best practices, idioms, and common pitfalls that can help you to write correct, understandable, and efficient C++ code. [Review]

• C++ Templates: The Complete Guide (David Vandevoorde and Nicolai M. Josuttis) This is the book about templates as they existed before C++11. It covers everything from the very basics to some of the most advanced template metaprogramming and explains every detail of how templates work (both conceptually and at how they are implemented) and discusses many common pitfalls. Has excellent summaries of the One Definition Rule (ODR) and overload resolution in the appendices. A second edition covering C++11, C++14 and C++17 has been already published. [Review]

• C++ 17 - The Complete Guide (Nicolai M. Josuttis) This book describes all the new features introduced in the C++17 Standard covering everything from the simple ones like 'Inline Variables', 'constexpr if' all the way up to 'Polymorphic Memory Resources' and 'New and Delete with over aligned Data'. [Review]

• C++ in Action (Bartosz Milewski). This book explains C++ and its features by building an application from ground up. [Review]

• Functional Programming in C++ (Ivan Čukić). This book introduces functional programming techniques to modern C++ (C++11 and later). A very nice read for those who want to apply functional programming paradigms to C++.

Advanced

• Modern C++ Design (Andrei Alexandrescu) A groundbreaking book on advanced generic programming techniques. Introduces policy-based design, type lists, and fundamental generic programming idioms then explains how many useful design patterns (including small object allocators, functors, factories, visitors, and multi-methods) can be implemented efficiently, modularly, and cleanly using generic programming. [Review]

• C++ Template Metaprogramming (David Abrahams and Aleksey Gurtovoy)

• C++ Concurrency In Action (Anthony Williams) A book covering C++11 concurrency support including the thread library, the atomics library, the C++ memory model, locks and mutexes, as well as issues of designing and debugging multithreaded applications. A second edition covering C++14 and C++17 has already been published. [Review]

• Advanced C++ Metaprogramming (Davide Di Gennaro) A pre-C++11 manual of TMP techniques, focused more on practice than theory. There are a ton of snippets in this book, some of which are made obsolete by type traits, but the techniques, are nonetheless useful to know. If you can put up with the quirky formatting/editing, it is easier to read than Alexandrescu, and arguably, more rewarding. For more experienced developers, there is a good chance that you may pick up something about a dark corner of C++ (a quirk) that usually only comes about through extensive experience.

• Large Scale C++ volume I, Process and architecture (John Lakos). Part one of a three-part series extending the older book 'Large Scale C++ Design'. Lakos explains battle-tested techniques to manage very big C++ software projects. If you work in a big C++ software project this is a great read, detailing the relationship between physical and logical structure, strategies for components, and their reuse.

Reference Style - All Levels

• The C++ Programming Language (Bjarne Stroustrup) (updated for C++11) The classic introduction to C++ by its creator. Written to parallel the classic K&R, this indeed reads very much like it and covers just about everything from the core language to the standard library, to programming paradigms to the language's philosophy. [Review] Note: All releases of the C++ standard are tracked in the question "Where do I find the current C or C++ standard documents?".

• C++ Standard Library Tutorial and Reference (Nicolai Josuttis) (updated for C++11) The introduction and reference for the C++ Standard Library. The second edition (released on April 9, 2012) covers C++11. [Review]

• The C++ IO Streams and Locales (Angelika Langer and Klaus Kreft) There's very little to say about this book except that if you want to know anything about streams and locales, then this is the one place to find definitive answers. [Review]

C++11/14/17/… References:

• The C++11/14/17 Standard (INCITS/ISO/IEC 14882:2011/2014/2017) This, of course, is the final arbiter of all that is or isn't C++. Be aware, however, that it is intended purely as a reference for experienced users willing to devote considerable time and effort to its understanding. The C++17 standard is released in electronic form for 198 Swiss Francs.

• The C++17 standard is available, but seemingly not in an economical form – directly from the ISO it costs 198 Swiss Francs (about $200 US). For most people, the final draft before standardization is more than adequate (and free). Many will prefer an even newer draft, documenting new features that are likely to be included in C++20.

• Overview of the New C++ (C++11/14) (PDF only) (Scott Meyers) (updated for C++14) These are the presentation materials (slides and some lecture notes) of a three-day training course offered by Scott Meyers, who's a highly respected author on C++. Even though the list of items is short, the quality is high.

• The C++ Core Guidelines (C++11/14/17/…) (edited by Bjarne Stroustrup and Herb Sutter) is an evolving online document consisting of a set of guidelines for using modern C++ well. The guidelines are focused on relatively higher-level issues, such as interfaces, resource management, memory management, and concurrency affecting application architecture and library design. The project was announced at CppCon'15 by Bjarne Stroustrup and others and welcomes contributions from the community. Most guidelines are supplemented with a rationale and examples as well as discussions of possible tool support. Many rules are designed specifically to be automatically checkable by static analysis tools.

• The C++ Super-FAQ (Marshall Cline, Bjarne Stroustrup, and others) is an effort by the Standard C++ Foundation to unify the C++ FAQs previously maintained individually by Marshall Cline and Bjarne Stroustrup and also incorporating new contributions. The items mostly address issues at an intermediate level and are often written with a humorous tone. Not all items might be fully up to date with the latest edition of the C++ standard yet.

• cppreference.com (C++03/11/14/17/…) (initiated by Nate Kohl) is a wiki that summarizes the basic core-language features and has extensive documentation of the C++ standard library. The documentation is very precise but is easier to read than the official standard document and provides better navigation due to its wiki nature. The project documents all versions of the C++ standard and the site allows filtering the display for a specific version. The project was presented by Nate Kohl at CppCon'14.

Classics / Older

Note: Some information contained within these books may not be up-to-date or no longer considered best practice.

• The Design and Evolution of C++ (Bjarne Stroustrup) If you want to know why the language is the way it is, this book is where you find answers. This covers everything before the standardization of C++.

• Ruminations on C++ - (Andrew Koenig and Barbara Moo) [Review]

• Advanced C++ Programming Styles and Idioms (James Coplien) A predecessor of the pattern movement, it describes many C++-specific “idioms”. It's certainly a very good book and might still be worth a read if you can spare the time, but quite old and not up-to-date with current C++.

• Large Scale C++ Software Design (John Lakos) Lakos explains techniques to manage very big C++ software projects. Certainly, a good read, if it only was up to date. It was written long before C++ 98 and misses on many features (e.g. namespaces) important for large-scale projects. If you need to work on a big C++ software project, you might want to read it, although you need to take more than a grain of salt with it. Not to be confused with the extended and later book series Large Scale C++ volume I-III.

• Inside the C++ Object Model (Stanley Lippman) If you want to know how virtual member functions are commonly implemented and how base objects are commonly laid out in memory in a multi-inheritance scenario, and how all this affects performance, this is where you will find thorough discussions of such topics.

• The Annotated C++ Reference Manual (Bjarne Stroustrup, Margaret A. Ellis) This book is quite outdated in the fact that it explores the 1989 C++ 2.0 version - Templates, exceptions, namespaces, and new casts were not yet introduced. Saying that however, this book goes through the entire C++ standard of the time explaining the rationale, the possible implementations, and features of the language. This is not a book to learn programming principles and patterns on C++, but to understand every aspect of the C++ language.

• Thinking in C++ (Bruce Eckel, 2nd Edition, 2000). Two volumes; is a tutorial-style free set of intro level books. Downloads: vol 1, vol 2. Unfortunately, they're marred by a number of trivial errors (e.g. maintaining that temporaries are automatic const), with no official errata list. A partial 3^rd party errata list is available at http://www.computersciencelab.com/Eckel.htm, but it is apparently not maintained.

• Scientific and Engineering C++: An Introduction to Advanced Techniques and Examples (John Barton and Lee Nackman) It is a comprehensive and very detailed book that tried to explain and make use of all the features available in C++, in the context of numerical methods. It introduced at the time several new techniques, such as the Curiously Recurring Template Pattern (CRTP, also called Barton-Nackman trick). It pioneered several techniques such as dimensional analysis and automatic differentiation. It came with a lot of compilable and useful code, ranging from an expression parser to a Lapack wrapper. The code is still available online. Unfortunately, the books have become somewhat outdated in the style and C++ features, however, it was an incredible tour-de-force at the time (1994, pre-STL). The chapters on dynamics inheritance are a bit complicated to understand and not very useful. An updated version of this classic book that includes move semantics and the lessons learned from the STL would be very nice.