Combination (unordered sample) is a method in combinatorics where the order does not matter. The opposite is permutations, where the order does matter.

 

When we form combinations, we distinguish between without replacement and with replacement.

 

 

Unordered sample without replacement

Imagine a deck of cards: How many different 5-card hands can you get from a set of 52 cards?

 

The sample is unordered, because the order in which the cards are drawn does not matter. It is also without replacement, because each card can only be used once.

 

$$ \large C(n,r) = \frac{n!}{(n-r)! \cdot r!} $$

$$ \large C(52,5) = \frac{52!}{(52-5)! \cdot 5!} $$

 

The result is 2,598,960 possible poker hands.

 

 

Unordered sample with replacement

If you must choose two letters from the set \(\{A, B, C, D, E\}\), and you are allowed to choose the same letter more than once, then this is an unordered sample with replacement.

 

This means that \(A,A\) is allowed, and that \(A,E\) is the same as \(E,A\), because the order does not matter.

 

$$ \large C(n,r) = \frac{(n-1+r)!}{(n-1)! \cdot r!} $$

$$ \large C(5,2) = \frac{(5-1+2)!}{(5-1)! \cdot 2!} = \frac{6!}{4! \cdot 2!} = \frac{720}{24 \cdot 2} = 15 $$

 

The 15 combinations are:

 

(A,A), (A,B), (A,C), (A,D), (A,E)
(B,B), (B,C), (B,D), (B,E)
(C,C), (C,D), (C,E)
(D,D), (D,E)
(E,E)

 

Note: In English textbooks the notation \(C(n,r)\) is commonly used. Another widely used form is the binomial coefficient:

 

$$ \large \binom{n}{r} $$

 

This is read as “n choose r”. It is the most universal notation and the one included in the formula collection.

 

 

Summary

A combination is an unordered sample, where the order does not matter.

• Without replacement: Each element can only be used once. Example: poker hands from 52 cards.
• With replacement: The same element can be chosen several times. Example: choosing letters where the same letter may appear more than once.

Combinations are used to count how many different ways a set of elements can be chosen when order does not matter.