How to Distinguish Repetition Allowed and Not Allowed in Permutations and Combinations

Introduction

When dealing with permutations and combinations, the concept of repetition is crucial in determining the correct formula to use. Permutations and combinations are mathematical concepts used to count the number of possible ways to arrange or select items from a set. In this article, we will explore how to identify situations where repetition is allowed or not allowed, and provide examples to clarify the concept.

Understanding Permutations

The primary difference between permutations and combinations lies in the order of the items and whether repetition is allowed. Let's delve into permutations first.

Permutations with Repetition Allowed

When repetition is allowed in permutations, it means that the same item can be used more than once.

The formula for permutations of n items taken r at a time with repetition is given by n^r. Example: Arranging the letters A, B, and C in 2 slots with repetition. The possible arrangements include AA, AB, AC, BA, BB, BC, CA, CB, and CC. Therefore, the number of arrangements is 3^2 9.

Permutations with Repetition Not Allowed

When repetition is not allowed, each item can only be used once.

The formula for permutations of n items taken r at a time without repetition is given by frac{n!}{(n - r)!}. Example: Arranging the letters A, B, and C in 2 slots without repetition. The possible arrangements include AB, AC, BA, BC, CA, and CB. Therefore, the number of arrangements is frac{3!}{(3 - 2)!} 6.

Understanding Combinations

Combinations are used when the order of the items does not matter, and repetition is an additional factor to consider.

Combinations with Repetition Allowed

In combinations with repetition allowed, you can choose the same item multiple times.

The formula for combinations of n items taken r at a time with repetition is given by binom{n r - 1}{r}. Example: Choosing 2 fruits from A, B, and C with repetition. The possible combinations include {A A}, {A B}, {A C}, {B B}, {B C}, and {C C}. Therefore, the number of combinations is binom{3 2 - 1}{2} 6.

Combinations with Repetition Not Allowed

In combinations with repetition not allowed, each item can only be chosen once.

The formula for combinations of n items taken r at a time without repetition is given by binom{n}{r} frac{n!}{r!(n - r)!}. Example: Choosing 2 fruits from A, B, and C without repetition. The possible combinations include {A B}, {A C}, and {B C}. Therefore, the number of combinations is binom{3}{2} 3.

Summary of Key Differences

To summarize:

Permutations: Order matters. When repetition is allowed, the formula is n^r; when not allowed, it is frac{n!}{(n - r)!}. Combinations: Order does not matter. When repetition is allowed, the formula is binom{n r - 1}{r}; when not allowed, it is binom{n}{r} frac{n!}{r!(n - r)!}.

Practical Application

Let's apply this knowledge to a practical problem. Consider the following scenario:

Permutation Problem Example: Arrange 3 fruits (A, B, and C) in 2 different slots.

Assume repetition is allowed. Possible arrangements are AA, AB, AC, BA, BB, BC, CA, CB, and CC. Therefore, the number of arrangements is 3^2 9. Assume repetition is not allowed. Possible arrangements are AB, AC, BA, BC, CA, and CB. Therefore, the number of arrangements is frac{3!}{(3 - 2)!} 6.

Note: If the question does not specify whether repetition is allowed or not, you should clearly state your assumption on the answer sheet. For example, if you assume repetition is allowed, write: "Since it is not specified, I am assuming repetition is allowed."