Specify order, replacement and geometry before reading the count. Main sizes are whole numbers from 0 to 10,000; zero selections have one arrangement. Fixed multiset frequencies are a separate linear, no-replacement model.
Read the result
Without replacement, rotations identify circular arrangements and reflections identify bracelets. With repeated colors, Burnside averages fixed strings over symmetry operations; dividing n^r by r or 2r is generally wrong because some strings have smaller orbits.
Counts use exact BigInt arithmetic. Scientific notation is a shortened display; Copy and CSV preserve exact integer text. Reciprocal-count probabilities require a uniform outcome model and do not establish password strength or real-world likelihood.
Conventions and limits
Pascal buttons transfer n and r without changing the counting mode, with undo. Binomial exponents range from 0 to 12: exponent zero gives 1. Compound bases are grouped, so (2x+3y)^2=(2x)^2+2·(2x)·(3y)+(3y)^2. Main Reset preserves these independent feature settings.
Check the counting model before interpreting probability
Name the objects, positions and equivalences before calculating. Linear ordered strings with replacement use n^r; distinct selections without replacement require r≤n. A fixed multiset instead specifies frequencies and counts arrangements by n!/(f₁!···fₖ!). It is a different model from choosing variable repeated colors.
Rotations identify circular arrangements; including reflections identifies bracelets. Exact counting does not make the outcomes equally likely. A probability of 1/count needs a uniform outcome model. Uniform raw strings need not produce uniform necklaces because orbit sizes can differ. Scientific notation is only a compact display; Copy and CSV preserve the exact integer.