This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The given letters are A, B, and C. So, the total number of distinct letters is n=3. We need to find the number of ordered arrangements when considering 2 letters at a time, so r=2. i) Replacement is allowed Step 1: Determine the number of choices for each position. Since replacement is allowed, for the first letter, there are 3 choices (A, B, or C). For the second letter, there are also 3 choices (A, B, or C), as the first letter can be repeated. Step 2: Calculate the total number of ordered arrangements. The number of ordered arrangements with replacement is given by n^r. 3^2 = 9 The possible arrangements are: AA, AB, AC, BA, BB, BC, CA, CB, CC. The number of ordered arrangements when replacement is allowed is 9. ii) Replacement is not allowed Step 1: Determine the number of choices for each position. Since replacement is not allowed, for the first letter, there are 3 choices (A, B, or C). For the second letter, there are only 2 remaining choices, as the first letter cannot be repeated. Step 2: Calculate the total number of ordered arrangements. The number of ordered arrangements without replacement is given by the permutation formula P(n, r) = (n!)/((n-r)!). P(3, 2) = (3!)/((3-2)!) = (3!)/(1!) = (3 × 2 × 1)/(1) = 6 The possible arrangements are: AB, AC, BA, BC, CA, CB. The number of ordered arrangements when replacement is not allowed is 6. Send me the next one 📸