MathematicsCombinatoricsA-Level

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Combinations (nCr)

Calculates the number of ways to choose 'r' items from a set of 'n' items without regard to the order of selection.

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^{n} C_{r} = (r n) = \frac{n !}{r ! ( n - r )!}

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Core idea

Overview

The combinations formula, denoted as nCr or (n choose r), determines how many distinct subsets of a given size r can be formed from a larger set of n distinct items. Unlike permutations, the order in which the items are chosen does not matter. This concept is fundamental in probability theory, statistics, and various fields requiring selection without regard to sequence, such as committee formation or lottery odds.

When to use: Use this formula when you need to count the number of ways to select a group of items from a larger collection, and the order of selection is not important. This is common in probability problems, experimental design, and situations where unique groupings are required.

Why it matters: Combinations are crucial for calculating probabilities in scenarios like card games, lotteries, and quality control, where the specific arrangement of chosen items is irrelevant. It helps in understanding the likelihood of events, designing experiments, and making informed decisions in fields ranging from genetics to computer science.

Symbols

Variables

n = Total Number of Items, r = Number of Items to Choose, {}^nC_r = Number of Combinations

n

Total Number of Items

i t e m s

r

Number of Items to Choose

i t e m s

^{n} C_{r}

Number of Combinations

co mbina t i o n s

Walkthrough

Derivation

Formula: Combinations (nCr)

The combinations formula calculates the number of ways to choose r items from n distinct items where the order of selection does not matter.

All n items are distinct.
The order of selection does not matter.
n and r are non-negative integers, and n ≥ r.

Start with Permutations:

The number of permutations (arrangements where order matters) of choosing r items from n is nPr. This counts ordered selections.

^{n} P_{r} = \frac{n !}{( n - r )!}

Account for Order (Divide by r!):

For every set of r chosen items, there are r! ways to arrange them. Since combinations disregard order, we divide the number of permutations by r! to remove the overcounting due to different orderings.

^{n} C_{r} = \frac{^{n} P _{r}}{r !}

Substitute Permutation Formula:

Substitute the formula for nPr into the expression for nCr.

^{n} C_{r} = \frac{\frac{n !}{( n - r )!}}{r !}

Simplify to Final Form:

This simplifies to the standard formula for combinations, also known as the binomial coefficient (n choose r).

^{n} C_{r} = \frac{n !}{r ! ( n - r )!}

Result

^{n} C_{r} = \frac{n !}{r ! ( n - r )!}

Source: AQA A-Level Mathematics — Statistics (7357/7356)

Visual intuition

Graph

Graph unavailable for this formula.

The graph of nCr plotted against r is a symmetric, bell-shaped polynomial curve that increases from (0, 1) to a maximum value at the midpoint (n/2) before decreasing symmetrically back to (n, 1). The shape reflects the property that the number of ways to choose r items is identical to the number of ways to choose the remaining n-r items, with the peak occurring at the point where the selection is most balanced. For larger values of n, this discrete distribution closely resembles a Gaussian or normal distribution curve.

Graph type: polynomial

Why it behaves this way

Intuition

Imagine a bag containing 'n' distinct balls. The formula counts the unique ways to reach into the bag and pull out 'r' balls, where the order you pull them out doesn't change the final set of balls you have.

The total count of distinct items available for selection.

A larger 'n' means more potential items to choose from, generally leading to more combinations for a fixed 'r'.

The specific number of items to be chosen from the total set.

The value of 'r' determines the size of each subset being formed; choosing 'r' items is different from choosing 'r+1' items.

Represents the total number of ordered arrangements (permutations) of all 'n' items.

This term initially overcounts by considering every possible sequence of all items, a starting point before adjusting for unordered selection.

Represents the number of ways to order the 'r' selected items among themselves.

Dividing by 'r!' corrects for the fact that the order of the chosen items does not matter in a combination; it removes the internal permutations of the selected group.

(n-r)!

Represents the number of ways to order the 'n-r' items that were not selected.

Dividing by '(n-r)!' corrects for the fact that the order of the unselected items also does not matter; it removes the internal permutations of the unselected group.

Free study cues

Insight

Canonical usage

This equation calculates a pure number representing a count of combinations, and thus has no physical units.

Common confusion

Students sometimes mistakenly try to assign units to 'n', 'r', or the result of the combination calculation, overlooking that these quantities represent counts of discrete items, which are pure numbers without physical

Dimension note

The quantities 'n' and 'r' represent counts of discrete items, which are inherently dimensionless non-negative integers. The result of the combination formula, the number of ways to choose 'r' items from 'n', is also a

Unit systems

$n$ dimensionless · Represents the total number of distinct items in the set, which is a count.

$r$ dimensionless · Represents the number of items to be chosen from the set, which is a count.

$n C r$ dimensionless · The resulting count of unique combinations, which is a pure number.

One free problem

Practice Problem

A committee of 3 people is to be chosen from a group of 8 candidates. How many different committees can be formed?

Total Number of Items8 items

Number of Items to Choose3 items

Solve for: $r es u l t$

Hint: Use the formula nCr = n! / (r!(n-r)!).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the number of possible lottery ticket combinations.

Study smarter

Tips

Remember that n must be greater than or equal to r (n ≥ r).
The formula involves factorials, so n! means n × (n-1) × ... × 1.
Combinations are about selection (order doesn't matter), while permutations are about arrangement (order matters).
For r=0 or r=n, nCr = 1 (there's one way to choose nothing or all items).

Avoid these traps

Common Mistakes

Confusing combinations with permutations (when order matters).
Incorrectly calculating factorials, especially for larger numbers.
Applying the formula when r > n (which is undefined).

Common questions

Frequently Asked Questions

The combinations formula calculates the number of ways to choose r items from n distinct items where the order of selection does not matter.

Use this formula when you need to count the number of ways to select a group of items from a larger collection, and the order of selection is not important. This is common in probability problems, experimental design, and situations where unique groupings are required.

Combinations are crucial for calculating probabilities in scenarios like card games, lotteries, and quality control, where the specific arrangement of chosen items is irrelevant. It helps in understanding the likelihood of events, designing experiments, and making informed decisions in fields ranging from genetics to computer science.

Confusing combinations with permutations (when order matters). Incorrectly calculating factorials, especially for larger numbers. Applying the formula when r > n (which is undefined).

Calculating the number of possible lottery ticket combinations.

Remember that n must be greater than or equal to r (n ≥ r). The formula involves factorials, so n! means n × (n-1) × ... × 1. Combinations are about selection (order doesn't matter), while permutations are about arrangement (order matters). For r=0 or r=n, nCr = 1 (there's one way to choose nothing or all items).

References

Sources

Wikipedia: Combination
Discrete Mathematics and Its Applications (Kenneth H. Rosen)
Discrete Mathematics and Its Applications, Kenneth H. Rosen
Wikipedia: Combination (mathematics)
Discrete Mathematics and Its Applications by Kenneth H. Rosen
A First Course in Probability by Sheldon M. Ross
AQA A-Level Mathematics — Statistics (7357/7356)