Engineering Mathematics · topic
Sum and product rules, permutations and combinations with and without repetition, distribution models, pigeonhole; inclusion-exclusion, derangements, Catalan and Stirling numbers; linear recurrences, homogeneous and non-homogeneous solution forms; ordinary generating functions and coefficient extraction.
Sum, product and complement rules, the four selection formulas (order and repetition), arrangements with repeated letters, circular arrangements, integer solutions with lower bounds, distributing distinct or identical objects into distinct or identical boxes (including onto distributions), lattice paths, the pigeonhole principle, binomial identities and the at-least-one overcounting trap, with every count checked by brute force.
A partial preview of the top of the sheet. The full sheet is in the Engineering Mathematics cheat sheet pack.
Inclusion-exclusion for two, three and n sets with a Venn diagram of region counts, counting integers not divisible by given primes and Euler's phi, solutions with upper bounds, derangements with three equivalent formulas and a past question with repeated letters, permutations with exactly k fixed points, Catalan numbers with the structures they count, Stirling numbers of the second kind and Bell numbers, and a brief note on Stirling numbers of the first kind.
A partial preview of the top of the sheet. The full sheet is in the Engineering Mathematics cheat sheet pack.
Setting up recurrences from counting problems, the characteristic equation for linear homogeneous recurrences with distinct and repeated roots, particular solutions for constant, polynomial and exponential right-hand sides including the resonance case, worked solutions checked with sympy, standard recurrences (Fibonacci, strings without 00, Hanoi, tilings), ordinary generating functions with a table of standard series, coefficient extraction, counting with restricted choices, and solving a recurrence through its generating function.
A partial preview of the top of the sheet. The full sheet is in the Engineering Mathematics cheat sheet pack.
