Combinations Calculator

Calculate combinations (nCr) with step-by-step solutions, Pascal's Triangle visualization, and practical examples like lottery odds and team selection.

Formula:C(n,r) = n! / (r! × (n-r)!)

Results

Combinations

120

C(10, 3)

Permutations P(n,r)720
10!3,628,800
Pascal's PositionRow 10, Position 3

Input Values

Total number of items to choose from

Number of items to select

CR(n,r) = C(n+r-1, r)

Visual representation (limited to n <= 12)

Results

Standard Combinations

Ways to select 3 items from 10 (order does not matter)

120

C(10, 3)

Combinations vs Permutations

Combinations (order does not matter)

120

Permutations (order matters)

720

Ratio: P(10,3) / C(10,3) = 6 = 3!

Step-by-Step Calculation

Step 1:C(10, 3) = 10! / (3! x (10 - 3)!)
Step 2:C(10, 3) = 10! / (3! x 7!)
Step 3:C(10, 3) = 3,628,800 / (6 x 5,040)
Step 4:C(10, 3) = 3,628,800 / 30,240
Step 5:C(10, 3) = 120

Pascal's Triangle

C(10, 3) = 120 is highlighted at row 10, position 3

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
252
210
120
45
10
1

Property: Sum of row 10 = 1,024 = 210

Practical Examples

Click to calculate:

Lottery Odds Calculator

Your Current Selection

Picking 3 numbers from 10

1 in 120

Probability: 0.83333333%

Common Lotteries

Powerball (5/69):1 in 11,238,513
Mega Millions (5/70):1 in 12,103,014
6/49 Lottery:1 in 13,983,816

Combination Formulas

Standard Combination

C(n, r) = n! / (r! x (n-r)!)

Order does not matter, no replacement

With Replacement

CR(n, r) = C(n+r-1, r)

Order does not matter, with replacement

Symmetry Property

C(n, r) = C(n, n-r)

C(10, 3) = C(10, 7)

Pascal's Identity

C(n, r) = C(n-1, r-1) + C(n-1, r)

Each Pascal entry is sum of two above

Results

Combinations

120

C(10, 3)

Permutations P(n,r)720

?How Do You Calculate Combinations?

Combinations count selections where order does NOT matter. Formula: C(n,r) = n! / (r! x (n-r)!). For example, choosing 3 people from 10 for a committee: C(10,3) = 10!/(3! x 7!) = 120 ways. With replacement, use CR(n,r) = C(n+r-1, r). Combinations are used for lottery odds, team selection, and any scenario where arrangement order is irrelevant.

What is a Combination?

A combination is a selection of items from a collection where the order of selection does not matter. Unlike permutations, combinations treat ABC and CBA as the same selection. The combination formula C(n,r) = n!/(r!(n-r)!) counts the number of ways to choose r items from n items. Combinations are fundamental to probability theory, statistics, and are commonly used in lottery calculations, committee formation, and sampling problems.

Key Facts About Combinations

  • Combination (nCr): order does NOT matter. Formula: C(n,r) = n!/(r!(n-r)!)
  • C(n,r) = C(n, n-r) - choosing r items is same as leaving (n-r) items
  • With replacement: CR(n,r) = C(n+r-1, r) allows repeated selections
  • nCr is always less than or equal to nPr (permutations)
  • Pascal's Triangle: each entry is C(row, position) = C(n,r)
  • Sum of row n in Pascal's Triangle equals 2^n
  • Lottery odds: C(49,6) = 13,983,816 combinations for 6 from 49
  • nC0 = nCn = 1 (one way to choose nothing or everything)

Try These Examples

Quick-start with common scenarios

Practice Mode

Test your skills with practice problems

Practice with 5 problems to test your understanding.

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Quick Answer

Combinations count selections where order does NOT matter. Formula: C(n,r) = n! / (r! x (n-r)!). For example, choosing 3 people from 10 for a committee: C(10,3) = 10!/(3! x 7!) = 120 ways. With replacement, use CR(n,r) = C(n+r-1, r). Combinations are used for lottery odds, team selection, and any scenario where arrangement order is irrelevant.

Frequently Asked Questions

A combination is a selection of items where ORDER DOES NOT MATTER. C(n,r) counts ways to choose r items from n items. Example: Choosing 3 people from 10 for a committee = C(10,3) = 120 different groups.

Permutation: ORDER MATTERS (ABC differs from BAC). Use for rankings, arrangements. Combination: ORDER DOES NOT MATTER (ABC = BAC). Use for selections, committees. P(n,r) is always >= C(n,r).

Combinations with replacement allow selecting the same item multiple times. Formula: CR(n,r) = C(n+r-1, r). Example: Choosing 3 scoops of ice cream from 5 flavors (can repeat) = CR(5,3) = C(7,3) = 35.

Pascal's Triangle is a triangular array where each number is the sum of the two numbers above it. The entry at row n, position r equals C(n,r). It's useful for quickly finding combination values and has many mathematical properties.

Lottery odds use combinations because order does not matter. For a 6/49 lottery: C(49,6) = 49!/(6! x 43!) = 13,983,816 possible combinations. Your odds of winning are 1 in 13,983,816.

Last updated: 2025-01-15