{"id":"integer-binomial-coefficient","version":"1.0.0","description":"Compute the exact binomial coefficient C(n, k), the number of ways to choose k items from n.","supported_operations":["integer binomial coefficient","n choose k","combinations count"],"unsupported_operations":["permutations count","multinomial coefficient","negative n or k"],"semantics":["n and k are JSON integers with 0 <= k <= n <= 100000.","The result is C(n, k) = n! / (k! * (n-k)!), computed exactly via an incremental multiply-then-exact-divide loop (never materializing n! itself), never floating point.","C(n, 0) and C(n, n) are 1 for every valid n.","Requests whose exact result would exceed 1000 digits are rejected rather than truncated."],"limits":{"max_n":100000,"max_digits":1000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["n","k"],"properties":{"n":{"type":"integer","minimum":0,"maximum":100000},"k":{"type":"integer","minimum":0,"maximum":100000}}},"output_schema":{"type":"object","additionalProperties":false,"required":["value"],"properties":{"value":{"type":"string"}}},"examples":[{"input":{"n":5,"k":2},"output":{"value":"10"}}],"execute_url":"/v1/tools/integer-binomial-coefficient/versions/1.0.0/execute"}