Description
Return an exact binomial factor for merging two category counts.
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 an exact binomial factor for merging two category counts.
When to use this tool
Use Multinomial Category Merge Factor when checking the stated word, set, multiset or composition convention.
How to use the tool and how it works
- Enter left, right. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read factor. Use the output convention below.
Return an exact binomial factor for merging two category counts. Require whole left/right in [0,128] and sum <= 256; factor = binomial(left+right,left) as decimal text.1
The exact binomial coefficient uses the shared calculation behind Pascal row; this helper returns only the local factor or summand described above.2
Multinomial counts divide the total factorial by multiplicity factorials; merging categories changes that denominator rather than preserving the arrangement count.3
| Input field | Meaning and units | Accepted input |
|---|---|---|
| left | Left | Required. Safe integer; minimum 0; maximum 128. |
| right | Right | Required. Safe integer; minimum 0; maximum 128. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| factor | Exact binomial(left+right,left): merged factorial denominator/separate denominator. Arrangement counts divide by this factor. | Required. Text; maximum UTF-16 length 8192. |
{
"input": {
"left": 2,
"right": 3
},
"output": {
"factor": "10"
}
}Limitations and assumptions
- Multiplying separate-category factorial denominators by this factor gives the merged denominator; the associated multinomial arrangement count divides by it.
- 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.
- Falling Factorial Last Factor Index
Return the last factor of a falling factorial with an empty convention.