Statistical & Engineering
Combinations Without & With Replacement Formula (COMBIN & COMBINA via FACT)
Calculate distinct k-item subsets without replacement C(n, k) = n! / (k!(n - k)!) and multisets with replacement C(n + k - 1, k) using FACT for audit sampling, portfolio pairs, and reliability redundancy.
Size audit samples, pairwise test matrices, and multiset allocations!
Choosing k distinct items from n candidates where order does NOT matter and items cannot repeat follows the binomial coefficient C(n, k) = n! / (k! * (n - k)!) (Excel’s COMBIN). For example, picking k = 3 servers out of n = 10 for failover drills yields 10! / (3! * 7!) = 120 distinct trios, whereas pairwise correlation matrices among n = 10 assets require C(10, 2) = 45 unique pairs.
When repetition IS allowed (stars-and-bars multiset selection, Excel’s COMBINA—such as allocating k = 3 identical compute credits across n = 10 microservices), the exact count expands to C(n + k - 1, k) = (n + k - 1)! / (k! * (n - 1)!) = 12! / (3! * 9!) = 220!
| Selection Rule (n = 10, k = 3) | Factorial Formula | Exact Count |
|---|---|---|
| Combinations Without Replacement (COMBIN) | 10! / (3! * 7!) | 120 |
| Combinations With Replacement (COMBINA) | 12! / (3! * 9!) | 220 |
| Ordered Permutations Without Replacement | 10! / 7! | 720 |
Combinations Without Replacement C(n, k)
Computes exact distinct k-element subsets from n items using FACT with integer and k <= n guards.
=IF(OR(B2 < 0, B3 < 0, B3 > B2), "Invalid n or k", ROUND(FACT(B2) / (FACT(B3) * FACT(B2 - B3)), 0))Multiset Combinations With Replacement C(n + k - 1, k)
Computes unordered selections of size k from n categories when repeats are permitted.
=IF(OR(B2 < 1, B3 < 0), "Invalid n or k", ROUND(FACT(B2 + B3 - 1) / (FACT(B3) * FACT(B2 - 1)), 0))Try the example
Use the original values or change them to test your understanding.