Math Problem Statement

how it was calculated

Find the permutation of the set A taking two elements at a time.

Solution

In the given example, the problem is to find the number of permutations of a set AA, which consists of 4 elements, taken 2 elements at a time.

Key Concepts:

  • Permutation Formula: The number of ways to select and arrange rr elements from a set of nn elements is given by: P(n,r)=n!(nr)!P(n, r) = \frac{n!}{(n - r)!} where n!n! denotes the factorial of nn, which is the product of all positive integers up to nn.

Given:

  • n=4n = 4 (since the set AA contains 4 elements: {1, 2, 3, 4}),
  • r=2r = 2 (we are choosing 2 elements at a time).

Step-by-Step Calculation:

  1. Apply the formula: P(4,2)=4!(42)!=4!2!P(4, 2) = \frac{4!}{(4 - 2)!} = \frac{4!}{2!}

  2. Evaluate the factorials:

    • 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24
    • 2!=2×1=22! = 2 \times 1 = 2
  3. Substitute and divide: P(4,2)=242=12P(4, 2) = \frac{24}{2} = 12

Thus, the number of permutations is 12. This means there are 12 ways to arrange 2 elements out of the 4-element set.

Would you like more details or have any questions about this calculation?

Here are 5 related questions to expand your understanding:

  1. How would you calculate the permutations if r=3r = 3 instead of 22?
  2. What is the difference between permutation and combination?
  3. How would you calculate combinations for the same values of nn and rr?
  4. What is the significance of the factorial function in permutation and combination formulas?
  5. Can you explain how the arrangement changes if we are not allowed to repeat elements?

Tip: For larger values of nn, using factorials directly can become tedious, so factorial tables or calculators are often used for quicker results.

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Math Problem Analysis

Mathematical Concepts

Permutation
Factorials

Formulas

Permutation formula P(n, r) = n! / (n - r)!

Theorems

Permutation Theorem

Suitable Grade Level

Grades 9-12