List all permutations of five objects a, b, c, d, and e taken three at a time without replacement.

Verified step by step guidance

1. Understand that a permutation of five objects taken three at a time means arranging 3 objects out of 5 in order, without repeating any object.
2. Recall the formula for the number of permutations of n objects taken r at a time: \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n=5\) and \(r=3\) in this case.
3. Calculate the total number of permutations using the formula: \(P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!}\), which gives the total count of different ordered arrangements.
4. List all possible ordered arrangements by selecting 3 objects from the set \{a, b, c, d, e\} without repetition, ensuring each arrangement is unique and order matters.
5. Organize the permutations systematically, for example, start with 'a' in the first position and list all pairs for the next two positions, then move to 'b' in the first position, and so on, until all permutations are listed.

Key Concepts

Permutations

Permutations refer to the arrangement of objects in a specific order. When order matters, each unique sequence counts as a different permutation. For example, arranging three letters from five distinct letters results in different permutations depending on the order.

Without Replacement

Without replacement means once an object is chosen, it cannot be selected again in the same arrangement. This affects the total number of possible permutations because each choice reduces the pool of available objects for subsequent positions.

Permutation Formula for n Objects Taken r at a Time

The number of permutations of n objects taken r at a time is calculated by n! / (n - r)!, where '!' denotes factorial. This formula counts all possible ordered arrangements of r objects selected from n distinct objects without replacement.