# Product Bundling Math: How Many Product Combos Can You Create?

n products make 2ⁿ − 1 possible bundles: 31 from 5 products, 1,023 from 10. Count fixed-size bundles with nCr, multiply across categories, then price them.

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- **Canonical URL:** https://dothecalculation.com/blog/business/product-bundling-math-how-many-combos
- **Category:** Business
- **Author:** Do The Calculation Team
- **Published:** 2026-10-07
- **Reading time:** 12 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

---

Product bundling means grouping products and selling them together for one price. Shops use it to raise the average order value, move slow stock alongside a bestseller, and give customers an easy "everything you need" choice.

Before you plan bundles, it helps to know how many are possible. The number grows faster than most people expect: 5 products can be grouped into 31 different bundles, and 10 products into 1,023. This guide shows the three formulas behind those numbers, works through examples, and ends with how to price a bundle so it still makes money.

In this guide:

- Why bundles are counted with combinations, not permutations
- Any-size bundles: the 2ⁿ − 1 formula
- Fixed-size bundles: the nCr formula
- One-from-each-category bundles: multiply the choices
- Bundles where a customer can repeat an item
- Reference tables, practice problems and pricing math

## Quick Answer

**Three ways to count bundles**
| Bundle type | Formula | Example with 5 products |
| --- | --- | --- |
| Any size, from 1 item to all of them | 2ⁿ − 1 | 2⁵ − 1 = 31 bundles |
| Exactly r items | nCr = n! ÷ (r! × (n − r)!) | 5C2 = 10 two-item bundles |
| One item from each category | Multiply the category sizes | 5 shirts × 3 hats = 15 outfits |

_[Figure: Possible any-size bundles as your catalogue grows - Number of products along the bottom. Each new product doubles the count and adds one: 2ⁿ − 1.]_

> **Key insight**: Any-size bundles grow exponentially: every extra product doubles the total. Fixed-size bundles grow much more slowly. With 10 products there are 1,023 bundles of any size but only 45 pairs.

## Why Bundles Are Combinations

A bundle of a shirt and a hat is the same bundle as a hat and a shirt. The order you list the items in does not matter, only which items are in the box. Counting problems where order does not matter use combinations. If order did matter (for example, ranking products 1st, 2nd and 3rd in an email), you would use permutations instead, and the numbers would be much larger. The [permutations vs combinations guide](/blog/math/permutations-vs-combinations) explains the difference in detail.

**The terms used below**
| Symbol | Meaning | Example |
| --- | --- | --- |
| n | How many distinct products you can choose from | 5 products |
| r | How many items go in each bundle | 2 items per bundle |
| nCr | Number of ways to choose r items from n, order ignored | 5C2 = 10 |
| 2ⁿ | Number of ways to include or leave out each of n products | 2⁵ = 32 |

## How to Count Any-Size Bundles (2ⁿ − 1)

**Any-size bundles**

```
Bundles = 2ⁿ − 1
```
- Each product has two options: in the bundle or out of it. With n products that is 2 × 2 × … × 2 = 2ⁿ choices.
- One of those choices leaves every product out. Subtract 1 to remove that empty "bundle".
- This count includes single items. Subtract n as well if a bundle must have at least 2 items: 2ⁿ − n − 1.

### Worked example 1: three products

With products A, B and C: 2³ − 1 = 8 − 1 = 7. Listing them confirms it: A, B, C, AB, AC, BC and ABC. If single items do not count as bundles, there are 7 − 3 = 4: AB, AC, BC and ABC.

### Worked example 2: five products, broken down by size

With 5 products, 2⁵ − 1 = 31. Those 31 bundles split by size into 5 singles, 10 pairs, 10 triples, 5 four-item bundles and 1 bundle of everything. Each size is a combination count: 5C1, 5C2, 5C3, 5C4 and 5C5.

_[Figure: The 31 bundles from 5 products, by bundle size - Middle sizes have the most options; the extremes have the fewest.]_

### Worked example 3: ten products

With 10 products, 2¹⁰ − 1 = 1,024 − 1 = 1,023 bundles. Ten products is a small catalogue, and it already gives more than a thousand possible bundles.

## How to Count Fixed-Size Bundles (nCr)

**Bundles of exactly r items**

```
nCr = n! ÷ (r! × (n − r)!)
```
- Shortcut: multiply r numbers counting down from n, then divide by r!.
- Example: 8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 336 ÷ 6 = 56.

### Worked example 4: two-item bundles from 5 products

5C2 = (5 × 4) ÷ (2 × 1) = 20 ÷ 2 = 10. The ten pairs are AB, AC, AD, AE, BC, BD, BE, CD, CE and DE.

### Worked example 5: three-item bundles from 8 products

8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 336 ÷ 6 = 56 bundles.

### Worked example 6: four-item bundles from 10 products

10C4 = (10 × 9 × 8 × 7) ÷ (4 × 3 × 2 × 1) = 5,040 ÷ 24 = 210 bundles.

_[Figure: Pairs vs triples as the catalogue grows - Number of products along the bottom. Fixed-size bundles grow steadily, not explosively; triples overtake pairs from 6 products.]_

Tool: [Count bundles with the Combination Calculator](https://dothecalculation.com/calculators/combination-calculator): Enter your number of products (n) and bundle size (r) to get nCr instantly.

## How to Count Bundles Across Categories

Many bundles take one item from each of several groups: a shirt, a pair of trousers and a hat, or a main, a side and a drink. Here you multiply, because every choice in one category can be paired with every choice in the next.

**One item from each category**

```
Bundles = (choices in category 1) × (choices in category 2) × …
```
- If a customer may also skip a category, add 1 to each category size for the "none" option, multiply, and subtract 1 for the empty order.

### Worked example 7: shirts and hats

5 shirts and 3 hats give 5 × 3 = 15 shirt-and-hat bundles. Adding them (5 + 3 = 8) is a common mistake: it counts items, not pairings.

### Worked example 8: a full outfit

4 shirts, 3 pairs of trousers and 2 hats give 4 × 3 × 2 = 24 complete outfits. If the hat is optional and so is everything else, it becomes (4 + 1) × (3 + 1) × (2 + 1) − 1 = 60 − 1 = 59 possible orders.

_[Figure: Which bundle formula do you need?]_

## Bundles Where Items Can Repeat

A "pick any 6" box of cookies, where a customer may choose three of the same flavour, is not covered by nCr, because nCr assumes every item in the bundle is different. When repeats are allowed and order does not matter, use (n + r − 1)Cr.

- A box of 6 cookies from 4 flavours, repeats allowed: (4 + 6 − 1)C6 = 9C6 = 84 boxes.
- A box of 3 candles from 6 scents, repeats allowed: 8C3 = 56 boxes. Without repeats it would be 6C3 = 20.

## Bundle Combination Tables

**Any-size bundles (2ⁿ − 1)**
| Products (n) | Any-size bundles |
| --- | --- |
| 3 | 7 |
| 4 | 15 |
| 5 | 31 |
| 6 | 63 |
| 7 | 127 |
| 8 | 255 |
| 9 | 511 |
| 10 | 1,023 |
| 15 | 32,767 |
| 20 | 1,048,575 |

**Fixed-size bundles (nCr)**
| n | r = 2 | r = 3 | r = 4 | r = 5 |
| --- | --- | --- | --- | --- |
| 5 | 10 | 10 | 5 | 1 |
| 6 | 15 | 20 | 15 | 6 |
| 7 | 21 | 35 | 35 | 21 |
| 8 | 28 | 56 | 70 | 56 |
| 9 | 36 | 84 | 126 | 126 |
| 10 | 45 | 120 | 210 | 252 |
| 12 | 66 | 220 | 495 | 792 |
| 15 | 105 | 455 | 1,365 | 3,003 |
| 20 | 190 | 1,140 | 4,845 | 15,504 |

## From Counting to Pricing: Does the Bundle Pay?

Knowing how many bundles exist tells you how much choice you have. Whether a bundle is worth offering depends on its margin. Take three products that sell for $25, $18 and $12 and cost you $10, $7 and $5. Sold separately they bring in $55 and cost $22, a $33 profit and a 60% margin.

**A 20% bundle discount, worked through**
|  | Sold separately | As a bundle (20% off) |
| --- | --- | --- |
| Price | $55.00 | $44.00 |
| Product cost | $22.00 | $22.00 |
| Profit | $33.00 | $22.00 |
| Margin | 60.0% | 50.0% |

_[Figure: Profit per order: what the customer would have bought vs the bundle - The bundle earns less than a full-price order of all three, but more than a customer who would only have bought the $25 item.]_

Run the same check on your own numbers with the [profit margin calculator](/calculators/profit-margin-calculator) and the [discount calculator](/calculators/discount-calculator). The [discount, markup and margin guide](/blog/business/discount-markup-strategy) explains why a discount takes a bigger bite out of margin than out of price. If a bundle mixes taxable and tax-exempt items, check your state's rules on bundled transactions too; the [sales tax guide for Shopify and Etsy sellers](/blog/business/sales-tax-guide-for-shopify-and-etsy-sellers) covers the basics of collecting tax online.

## Practice Problems (With Answers)

**Try these, then check the working**
| Problem | Working | Answer |
| --- | --- | --- |
| 1. 4 products, any-size bundles | 2⁴ − 1 = 16 − 1 | 15 |
| 2. 6 products, 2-item bundles | 6C2 = (6 × 5) ÷ 2 | 15 |
| 3. 7 products, any-size bundles | 2⁷ − 1 = 128 − 1 | 127 |
| 4. 8 products, 3-item bundles | 8C3 = 336 ÷ 6 | 56 |
| 5. 4 shirts, 3 trousers, 2 hats, one of each | 4 × 3 × 2 | 24 |
| 6. 10 products, 4-item bundles | 10C4 = 5,040 ÷ 24 | 210 |
| 7. 8 products, bundles of 2 to 4 items | 8C2 + 8C3 + 8C4 = 28 + 56 + 70 | 154 |

For more counting practice with full step-by-step solutions, work through the [permutation and combination practice problems](/blog/math/permutation-and-combination-practice-problems).

## Common Bundling Math Mistakes

**Mistakes and fixes**
| Mistake | Why it is wrong | Fix |
| --- | --- | --- |
| Using permutations | Counts AB and BA as two bundles | Bundles ignore order: use nCr |
| Not subtracting 1 from 2ⁿ | Counts the empty bundle | Use 2ⁿ − 1 |
| Adding category sizes | 5 shirts + 3 hats = 8 counts items, not pairs | Multiply: 5 × 3 = 15 |
| Using 2ⁿ − 1 for a fixed size | It counts every size at once | Use nCr for exactly r items |
| Using nCr when repeats are allowed | nCr assumes all items differ | Use (n + r − 1)Cr |
| Counting near-duplicates as distinct | Two colours of one item may not be a real choice | Decide what counts as a different product first |
| Offering every possible bundle | Too much choice and too many SKUs to manage | Test a few bundles built around your best sellers |

## Frequently Asked Questions

**How many bundles can I create with 5 products?**

31 bundles of any size (2⁵ − 1), of which 10 are pairs and 10 are three-item bundles. If single items do not count, there are 26.

**What is the formula for any-size bundles?**

2ⁿ − 1, where n is the number of products. Each product is either in or out (2 options), and the 1 removes the empty bundle.

**What is the formula for bundles of a fixed size?**

nCr = n! ÷ (r! × (n − r)!), where n is the number of products and r is the number of items per bundle.

**Why use combinations and not permutations for bundles?**

A bundle is defined by which items are in it, not the order they are listed in. AB and BA are the same bundle, so order is ignored.

**How do I count bundles across categories?**

Multiply the number of choices in each category. 5 shirts and 3 hats give 15 shirt-and-hat bundles.

**How should I price a bundle?**

Price it below the sum of the separate prices so customers see a saving, then check the margin. A 20% discount on items with a 60% margin leaves 50%: the $11 discount comes straight out of the $33 profit. A bundle pays when it turns single-item buyers into multi-item buyers.

## Final Summary

- Any-size bundles: 2ⁿ − 1 (31 from 5 products, 1,023 from 10).
- Exactly r items: nCr (10 pairs from 5 products).
- One from each category: multiply the category sizes.
- Repeats allowed: (n + r − 1)Cr.
- Count first, then check each bundle's margin before you offer it.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/business/product-bundling-math-how-many-combos). Quote freely with attribution and a link to this page._