Description
Return the last factor of a falling factorial with an empty convention.
Combinatorial counts depend on whether objects are labeled, order matters, repetitions are permitted and a prefix is already fixed. Local counts, exact integer factors and feasibility bounds answer specific questions and need not enumerate complete objects.
Return the last factor of a falling factorial with an empty convention.
When to use this tool
Use Falling Factorial Last Factor Index when checking the stated word, set, multiset or composition convention.
How to use the tool and how it works
- Enter n, k. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read lastFactor, empty. Inspect empty before interpreting the associated values.
Return the last factor of a falling factorial with an empty convention. Require valid whole n and k in [0,n]; lastFactor = n-k+1 when k > 0, otherwise one with empty true.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| n | N | Required. Safe integer; minimum 0. |
| k | K | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| lastFactor | One for an empty product, otherwise n-k+1; this is not the complete falling factorial. | Required. Safe integer. |
| empty | Empty | Required. Boolean. |
{
"input": {
"n": 7,
"k": 3
},
"output": {
"lastFactor": 5,
"empty": false
}
}Limitations and assumptions
- The empty result is a multiplicative sentinel, not a literal final factor; the whole falling factorial is not evaluated.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use complete objects and an appropriate exact counting, ranking or enumeration method when the local contribution is insufficient.
References
Similar or alternative tools
- Combinatorics: Multiset Multiplicity Factorial Denominator
Return the exact product of multiplicity factorials.
- Multinomial Category Merge Factor
Return an exact binomial factor for merging two category counts.