Solve permutation, combination, and constraint satisfaction problems. Calculate possible arrangements, team selections, scheduling scenarios, and probability-based outcomes. This calculator helps solve real-world discrete math problems.
Select Problem Type
Permutation Calculator: P(n, r)
Combination Calculator: C(n, r)
Team Selection with Constraints
Scheduling & Assignment Problem
Introduction: why Combinatorics & Constraint Solver matters
In the real world, the hard part is rarely finding a formula—it is turning a messy situation into a small set of inputs you can measure, validating that the inputs make sense, and then interpreting the result in a way that leads to a better decision. That is exactly what a calculator like Combinatorics & Constraint Solver is for. It compresses a repeatable process into a short, checkable workflow: you enter the facts you know, the calculator applies a consistent set of assumptions, and you receive an estimate you can act on.
People typically reach for a calculator when the stakes are high enough that guessing feels risky, but not high enough to justify a full spreadsheet or specialist consultation. That is why a good on-page explanation is as important as the math: the explanation clarifies what each input represents, which units to use, how the calculation is performed, and where the edges of the model are. Without that context, two users can enter different interpretations of the same input and get results that appear wrong, even though the formula behaved exactly as written.
This article introduces the practical problem this calculator addresses, explains the computation structure, and shows how to sanity-check the output. You will also see a worked example and a comparison table to highlight sensitivity—how much the result changes when one input changes. Finally, it ends with limitations and assumptions, because every model is an approximation.
What problem does this calculator solve?
The underlying question behind Combinatorics & Constraint Solver is usually a tradeoff between inputs you control and outcomes you care about. In practice, that might mean cost versus performance, speed versus accuracy, short-term convenience versus long-term risk, or capacity versus demand. The calculator provides a structured way to translate that tradeoff into numbers so you can compare scenarios consistently.
Before you start, define your decision in one sentence. Examples include: “How much do I need?”, “How long will this last?”, “What is the deadline?”, “What’s a safe range for this parameter?”, or “What happens to the output if I change one input?” When you can state the question clearly, you can tell whether the inputs you plan to enter map to the decision you want to make.
How to use this calculator
- Select a problem type from the tabs: Permutation, Combination, Team Selection, or Scheduling.
- For permutations, enter the total number of items (n) and items to arrange (r). Order matters here.
- For combinations, enter total items and items to choose. Order does not matter.
- For team selection, specify team size, required roles, and any constraints on who must or cannot be included.
- For scheduling, define time slots, participants, and any forbidden pairings.
Results show the count of valid arrangements and, where applicable, a breakdown of the solution space.
Inputs: how to pick good values
The calculator’s form collects the variables that drive the result. Many errors come from unit mismatches (hours vs. minutes, kW vs. W, monthly vs. annual) or from entering values outside a realistic range. Use the following checklist as you enter your values:
- Units: confirm the unit shown next to the input and keep your data consistent.
- Ranges: if an input has a minimum or maximum, treat it as the model’s safe operating range.
- Defaults: defaults are example values, not recommendations; replace them with your own.
- Consistency: if two inputs describe related quantities, make sure they don’t contradict each other.
Key parameters for combinatorics problems:
- n (total items): The size of the set you are selecting from.
- r (items to select): How many items you want in each arrangement or combination.
- Required items: Items that must appear in every valid selection.
- Forbidden items: Items that cannot appear together in the same selection.
If you are unsure about a value, it is better to start with a conservative estimate and then run a second scenario with an aggressive estimate. That gives you a bounded range rather than a single number you might over-trust.
Formulas: how the calculator turns inputs into results
Most calculators follow a simple structure: gather inputs, normalize units, apply a formula or algorithm, and then present the output in a human-friendly way. Even when the domain is complex, the computation often reduces to combining inputs through addition, multiplication by conversion factors, and a small number of conditional rules.
At a high level, you can think of the calculator’s result R as a function of the inputs x 1 … x n :
A very common special case is a “total” that sums contributions from multiple components, sometimes after scaling each component by a factor:
Here, w i represents a conversion factor, weighting, or efficiency term. That is how calculators encode “this part matters more” or “some input is not perfectly efficient.” When you read the result, ask: does the output scale the way you expect if you double one major input? If not, revisit units and assumptions.
Worked example (step-by-step)
Worked examples are a fast way to validate that you understand the inputs. For illustration, suppose you enter the following three values:
- Input 1: 1
- Input 2: 2
- Input 3: 3
A simple sanity-check total (not necessarily the final output) is the sum of the main drivers:
Sanity-check total: 1 + 2 + 3 = 6
After you click calculate, compare the result panel to your expectations. If the output is wildly different, check whether the calculator expects a rate (per hour) but you entered a total (per day), or vice versa. If the result seems plausible, move on to scenario testing: adjust one input at a time and verify that the output moves in the direction you expect.
Comparison table: sensitivity to a key input
The table below changes only Input 1 while keeping the other example values constant. The “scenario total” is shown as a simple comparison metric so you can see sensitivity at a glance.
| Scenario | Input 1 | Other inputs | Scenario total (comparison metric) | Interpretation |
|---|---|---|---|---|
| Conservative (-20%) | 0.8 | Unchanged | 5.8 | Lower inputs typically reduce the output or requirement, depending on the model. |
| Baseline | 1 | Unchanged | 6 | Use this as your reference scenario. |
| Aggressive (+20%) | 1.2 | Unchanged | 6.2 | Higher inputs typically increase the output or cost/risk in proportional models. |
In your own work, replace this simple comparison metric with the calculator’s real output. The workflow stays the same: pick a baseline scenario, create a conservative and aggressive variant, and decide which inputs are worth improving because they move the result the most.
How to interpret the result
The results panel is designed to be a clear summary rather than a raw dump of intermediate values. When you get a number, ask three questions: (1) does the unit match what I need to decide? (2) is the magnitude plausible given my inputs? (3) if I tweak a major input, does the output respond in the expected direction? If you can answer “yes” to all three, you can treat the output as a useful estimate.
When relevant, a CSV download option provides a portable record of the scenario you just evaluated. Saving that CSV helps you compare multiple runs, share assumptions with teammates, and document decision-making. It also reduces rework because you can reproduce a scenario later with the same inputs.
Limitations and assumptions
No calculator can capture every real-world detail. This tool aims for a practical balance: enough realism to guide decisions, but not so much complexity that it becomes difficult to use. Keep these common limitations in mind:
- Input interpretation: the model assumes each input means what its label says; if you interpret it differently, results can mislead.
- Unit conversions: convert source data carefully before entering values.
- Linearity: quick estimators often assume proportional relationships; real systems can be nonlinear once constraints appear.
- Rounding: displayed values may be rounded; small differences are normal.
- Missing factors: local rules, edge cases, and uncommon scenarios may not be represented.
If you use the output for compliance, safety, medical, legal, or financial decisions, treat it as a starting point and confirm with authoritative sources. The best use of a calculator is to make your thinking explicit: you can see which assumptions drive the result, change them transparently, and communicate the logic clearly.
Definitions (n, r, combinations vs permutations)
n is the total number of distinct items available. r is how many items you select.
- Combinations (nCr) : order does not matter (e.g., choosing 3 people for a committee).
- Permutations (nPr) : order matters (e.g., assigning 1st/2nd/3rd place).
How the constraints work on this page
This calculator applies two count-based constraints:
- Required count (k) : assumes there are exactly k “required” items and they must all be included in the selection. It then counts ways to choose the remaining r − k items from the other n − k items.
- Forbidden count (f) : removes f items from the pool and counts outcomes from the remaining n − f items.
Important: in the current implementation, if you enter both required and forbidden counts, the forbidden step is applied afterward and effectively overrides the earlier “required” adjustment rather than combining both constraints. Use one constraint at a time unless you update the logic.
Edge cases & input rules
- If r > n , there are no valid outcomes.
- If required count > r or required count > n , there are no valid outcomes.
- If forbidden count makes n − forbidden < r , there are no valid outcomes.
- Inputs are treated as non-negative integers ; repeated items are not supported.
How the constraint math works
Base count (no constraints)
- Combinations (nCr) : order does not matter.
- Permutations (nPr) : order matters.
Required items (count only)
If
k
items are required, they must appear in the selection. The calculator effectively “locks in” those
k
items and counts ways to choose/arrange the remaining
r − k
items from the other
n − k
items.
-
Combinations:
C(n − k, r − k) -
Permutations (as implemented here):
P(n − k, r − k)for the remaining slots
Forbidden items (count only)
If
f
items are forbidden, they are removed from the pool first, leaving
n − f
allowed items.
-
Combinations:
C(n − f, r) -
Permutations:
P(n − f, r)
When both required and forbidden are entered
This calculator applies the required-item adjustment and then (if any forbidden items are entered) recalculates using only the forbidden adjustment. If you need “required and forbidden simultaneously” (i.e., required items are a specific subset disjoint from forbidden items), the combined form is typically:
-
Combinations:
C(n − k − f, r − k)(when inputs are consistent) - Permutations: depends on whether order among required items is also counted
How the constrained nCr/nPr calculation works
This calculator counts how many ways to select r items from n items when:
- k items are required (must be included), and/or
- f items are forbidden (cannot be used).
It assumes all items are distinct and constraints are specified as counts (not specific identities).
Formulas used
- Combinations (order doesn’t matter): valid = C(n − f − k, r − k)
- Permutations (order matters): valid = P(n − f − k, r − k)
If k = 0 , this reduces to C(n − f, r) or P(n − f, r).
FAQ
When should I use combinations vs permutations?
Use combinations when order doesn’t matter (a committee). Use permutations when order matters (ranked positions, seat assignments, codes).
Why does the result become 0?
Because the constraints make the selection impossible (for example, k > r, or forbidding leaves fewer than r allowed items).
Does this handle “at least one from group A” style rules?
No. This page supports only “exactly k required” (must include all k) and “exactly f forbidden” (exclude all f) as counts. Group rules typically require inclusion–exclusion.
How this constrained nCr/nPr calculator works
This calculator counts how many selections of size r you can make from n distinct items when:
- k required items must be included, and/or
- f forbidden items are not allowed to appear.
Choose Combination (nCr) when order does not matter, or Permutation (nPr) when order matters.
Inputs (definitions & valid ranges)
- Total items (n): number of distinct available items.
- Selection size (r): how many items are chosen.
- Required items (k): how many specific items must be included (0 to min(n, r)).
- Forbidden items (f): how many specific items cannot be used (0 to n).
- Mode: combination (unordered) or permutation (ordered).
Formula used (required/forbidden constraints)
Let allowed = n − f be the number of items remaining after removing forbidden items.
-
If
k = 0
:
- Combinations: C(allowed, r)
- Permutations: P(allowed, r)
-
If
k > 0
: you must include those k required items, so you only choose the remaining
r − k
items from the
allowed − k
non-required allowed items:
- Combinations: C(allowed − k, r − k)
- Permutations: P(allowed − k, r − k) (counts ways to arrange the non-required portion as implemented on this page)
Outputs show the constrained total and the unconstrained “base” nCr/nPr for comparison.
Common edge cases & sanity checks
- If r > n , there are no valid selections.
- If k > r (you require more items than you select), result is 0.
- If allowed < r (too many forbidden items), result is 0.
- As constraints increase (larger k or f), the constrained total should not increase.
How the constraints change the count
This calculator counts the number of valid selections when you start with n total items, choose r of them, and apply two types of constraints:
- Forbidden items (f) : items that cannot be chosen. This reduces the pool from n to n − f .
- Required items (k) : items that must be included in the selection. Once required items are fixed, you only need to choose the remaining r − k items from the remaining pool.
Combinations (order does not matter)
If order doesn’t matter (nCr), the count is:
- No required items: C(n − f, r)
- With required items: C((n − f) − k, r − k)
Permutations (order matters)
If order matters (nPr), the count is:
- No required items: P(n − f, r)
- With required items: P((n − f) − k, r − k)
Note: This tool treats “required” and “forbidden” as counts of distinct items (not specific identities). If you need exact items (e.g., “Alice is required”), the math is the same as long as the items are distinct.
What this calculator computes (required & forbidden items)
This calculator counts how many ways you can choose r items from n total items when:
- Required count (k) : exactly k specific items must be included in every outcome.
- Forbidden count (f) : exactly f specific items are not allowed to appear in any outcome.
You can switch between:
- Combinations (nCr) : order does not matter.
- Permutations (nPr) : order does matter.
How the calculation works (high level)
Let allowed = n − f be the number of items that remain after excluding forbidden items.
- If you have k required items, they are “pre-selected”, leaving r − k remaining picks to make from allowed − k items.
- If you have no required items, you pick all r from allowed .
The result is:
- Combinations : C(allowed − k, r − k) or C(allowed, r)
- Permutations : P(allowed − k, r − k) or P(allowed, r)
Input rules & sanity checks
- All inputs must be non-negative integers.
- If r > n , there are no valid outcomes.
- If k > r (more required than you’re selecting), the answer is 0 .
- If f > n , the answer is 0 .
- If k + f > n , the constraints cannot be satisfied (the answer is 0 ).
- If allowed < r (not enough non-forbidden items to fill the selection), the answer is 0 .
- Increasing f should never increase the count; increasing k generally reduces flexibility and should not increase the count.