Variations in Combinatorics

INTRODUCTION TO ORDERED SELECTIONS

In combinatorics, variations are used to count the possible ways to arrange a chosen number of elements from a larger set where the order is important. This approach is crucial for tasks like ranking, coding, creating passwords, or seating people, especially when it matters who is first, second, third, and so on.

With variations, it's not necessary to use all elements of the set. From a set with n elements, we choose k elements (where k ≤ n) and arrange them in an ordered sequence.

VARIATIONS WITHOUT REPETITION

When we do not allow the repetition of elements, we are talking about variations without repetition. The number of these variations is denoted by V(n, k) and is calculated using the formula: V(n, k) = n * (n – 1) * (n – 2) * … * (n – k + 1)

Or, written with factorials: V(n, k) = n! / (n – k)!

EXAMPLE: How many different two-digit numbers can be made using the digits 1 through 5, without repetition?

• n = 5 (number of available digits)
• k = 2 (we are forming two-digit numbers) V(5, 2) = 5 * 4 = 20

VARIATIONS WITH REPETITION

If repetition is allowed, then each of the k positions can be filled with any of the n elements, regardless of what has already been chosen.

The formula for variations with repetition is: V'(n, k) = n^k

EXAMPLE: How many different two-digit numbers can be made using the digits 1 through 5 if repetition is allowed?

• n = 5
• k = 2 V'(5, 2) = 5^2 = 25

DIFFERENCE BETWEEN VARIATIONS AND PERMUTATIONS

• In permutations, we arrange all the elements (k = n).
• In variations, we arrange only a subset of the elements (k < n).
• In both cases, the order is important.

EXAMPLE: Given the set {A, B, C}, so n = 3:

• Permutations (k = 3): ABC, ACB, BAC, BCA, CAB, CBA → 3! = 6
• Variations without repetition (k = 2): AB, AC, BA, BC, CA, CB → V(3, 2) = 6

PRACTICAL USES

Variations appear in various contexts:

• Determining possible passwords or codes.
• Ranking finalists based on their finishing positions.
• Forming combinations of cards, numbers, or symbols where order plays a role.

CONCLUSION

Variations are a fundamental tool in combinatorics for handling the ordered selection of a smaller number of elements from a given set. We distinguish between cases with and without repetition, but in every scenario, the key takeaway is that the order of the selected elements is not negligible and directly impacts the final result.