Combinatorics

• Asymptotics of the Central Binomial Coefficient
  Using Stirling’s formula, the central binomial coefficient satisfies C(2n,n) ~ 4^n divided by √(πn), with useful correction terms.
• Why You Cannot Connect 3 Dots to 3 Squares Without Crossings
  The three-dot, three-square puzzle requires drawing K3,3. Euler’s formula shows it needs nine edges, but a planar bipartite graph on six vertices allows at most eight.
• Examples of Open Problems Solved by Short Proofs
  See examples where a compact argument settled a mathematical problem, and learn why a short proof can hide a long search or earlier work.
• How to Distribute Identical Stones Among Buckets with Limits
  Count distributions of identical stones into distinct buckets with minimum and maximum limits using stars and bars, inclusion-exclusion, or generating functions.
• How to Calculate Probabilities in Minesweeper
  Calculate a Minesweeper square’s mine probability by counting all mine layouts consistent with clues and the total mine count.
• How Many Ways Can Men and Women Sit Around a Table with No Adjacent Men?
  For n distinct men and k distinct women seated up to rotation, count the arrangements by placing women first and filling separate gaps.
• How to Count Integers Not Divisible by 5
  To count integers not divisible by 5, subtract the multiples of 5 from the total; use floor formulas for any interval or digit rule.
• How to Find a Closed Form for an nth Term Using Generating Functions
  Turn a recurrence into a generating function, simplify it, and extract its coefficients to obtain a closed form for the nth term.
• How Many Ways Are There to Eat Lunch?
  There is no single count without a menu and rules. Define the lunch choices, then use the product rule, sum rule, or permutations as needed.
• How to Find Probabilities in a Box of Red and Black Balls
  For a draw without replacement, use combinations to find the probability of a given number of red and black balls, or adapt the method for replacement.
• Can 7 Drinks with 7 Flavors Make Infinite Varieties?
  With continuously variable proportions, seven flavors produce uncountably many mathematical mixtures; counting only flavor subsets gives 127 nonempty choices.
• How to Solve the Missionaries and Cannibals Problem
  Solve the classic three-missionary, three-cannibal river crossing in 11 boat trips, with a step-by-step table and an explanation of the safety rule.
• Can You Draw This Figure Without Lifting the Pen? (Euler Paths)
  A figure can be drawn in one stroke exactly when it is connected and has 0 or 2 points where an odd number of lines meet. How to check, and where to start.