Determine whether the following problem involves a permutation or a combination and explain your answer How many different 3-letter passwords can be formed from the letters A, B, C, D, E, F, and G if no repetition of letters is allowed? Choose the correct answer below. A. The problem involves a combination because the order in which the letters are selected does matter. B. The problem involves a combination because the order in which the letters are selected does not matter. C. The problem involves a permutation because the order in which the letters are selected does not matter. D. The problem involves a permutation because the order in which the letters are selected does matter.
Solution
Step 1 ：The problem is asking for the number of different 3-letter passwords that can be formed from the given letters without repetition. This means that the order in which the letters are selected does matter. For example, 'ABC' and 'CBA' would be considered different passwords. Therefore, this problem involves a permutation, not a combination.
Step 2 ：We can calculate the number of permutations using the formula \(P(n, r) = \frac{n!}{(n-r)!}\), where n is the number of items to choose from (in this case, 7 letters), and r is the number of items to choose (in this case, 3 letters).
Step 3 ：Substituting the given values into the formula, we get \(P(7, 3) = \frac{7!}{(7-3)!} = 210\).
Step 4 ：Final Answer: The correct answer is D. The problem involves a permutation because the order in which the letters are selected does matter. The number of different 3-letter passwords that can be formed from the letters A, B, C, D, E, F, and G without repetition is \(\boxed{210}\).
