Combination Calculator With Replacement

The calculator uses the standard combination with replacement formula:

C(n + r - 1, r)

Here, n is the number of available item types. The value r is the number of selections made. Replacement means an item type can be selected more than once. Order does not matter.

The factorial form is:

C(n + r - 1, r) = (n + r - 1)! / (r! × (n - 1)!)

This page also compares the answer with normal combinations and ordered replacement choices. Normal combinations use C(n, r). Ordered replacement choices use n^r.

1. Enter the number of unique item types in the first field.
2. Enter how many selections you want to make.
3. Add optional item labels if you want sample combinations.
4. Set a sample limit for small cases.
5. Press Calculate to view the result below the header.
6. Use the CSV or PDF buttons to export the calculation.

What This Count Means

A combination with replacement counts selections where repeats are allowed. Order is ignored. This makes it different from a sequence. It also differs from a normal combination, where each item can be used only once.

Think about choosing scoops of ice cream. You may choose vanilla twice. You may also choose chocolate once. The order of scoops does not change the final group. Vanilla, vanilla, chocolate is the same group as chocolate, vanilla, vanilla.

Why the Formula Works

The formula is based on a stars and bars idea. The selected items are the stars. The dividers separate item types. If there are n item types and r selections, the problem becomes arranging r stars with n minus one dividers. That gives C(n + r - 1, r).

This method works because each arrangement maps to one repeated selection pattern. For example, three stars before the first divider means three copies of the first item. A blank section means zero copies of that item.

When to Use It

Use this calculator for menu choices, test cases, product bundles, card ranks, coins, dice groups, classroom examples, and inventory planning. It is useful whenever repeated choices are valid and order is not important.

The comparison metrics help you avoid common mistakes. If order matters, use ordered replacement instead. If repeats are not allowed, use combinations without replacement. These counts can be very different.

Large Result Handling

Combination values grow fast. Small inputs may produce large totals. This calculator uses exact integer steps when practical. For very large inputs, it also provides a scientific estimate and digit count. These outputs are helpful for reports and probability checks.

The sample list is limited on purpose. Listing every combination can become huge. The count is usually more useful than the complete list. Export options make it easier to save results for lessons, worksheets, and planning records.