1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.

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Understand the concept of permutations: Permutations refer to the arrangement of objects in a specific order. When calculating the number of permutations of n distinct objects taken r at a time, you are counting the number of possible ordered arrangements of r objects selected from a total of n objects.

Recall the formula for permutations: The number of permutations of n distinct objects taken r at a time is given by the formula P(n, r) = n! / (n - r)!, where '!' denotes factorial (the product of all positive integers up to that number).

Break down the formula: The numerator, n!, represents the total number of ways to arrange all n objects. The denominator, (n - r)!, removes the arrangements of the remaining (n - r) objects that are not selected, leaving only the arrangements of the r selected objects.

Provide an example: Suppose you have 5 distinct objects (A, B, C, D, E) and want to calculate the number of permutations when selecting 3 objects at a time. Using the formula, P(5, 3) = 5! / (5 - 3)!. This simplifies to 5! / 2!.

Explain the result conceptually: The result represents the total number of ways to arrange 3 objects (in a specific order) chosen from the 5 distinct objects. For example, one possible arrangement is (A, B, C), and another is (C, A, B).

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주요 개념

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Permutations

Permutations refer to the different arrangements of a set of objects where the order matters. When calculating permutations of n distinct objects taken r at a time, you are counting the number of ways to arrange r objects selected from a total of n. For example, if you have three distinct letters A, B, and C, the permutations of 2 letters taken from these would include AB, AC, BA, BC, CA, and CB, totaling six arrangements.

Factorial

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For instance, 4! equals 4 × 3 × 2 × 1 = 24, which represents the total arrangements of four distinct objects.

Combination vs. Permutation

While both permutations and combinations deal with selecting items from a set, the key difference lies in the importance of order. Permutations consider the arrangement of selected items as significant, whereas combinations focus solely on the selection itself, disregarding order. For example, selecting the letters A and B is the same in combinations (AB) but different in permutations (AB vs. BA).