Final answer:
To find the different arrangements of 6 letters using 3 from 'article' and 3 from 'show', calculate the combinations for each selection and then the permutations. There are 35 combinations from 'article', 4 from 'show', and 720 permutations for a total of 100,800 different arrangements.
Step-by-step explanation:
To determine how many different arrangements of 6 letters are possible using 3 letters from "article" and 3 letters from "show," we must consider the combinations and permutations of the letters chosen. Let's start with the word "article." There are 7 letters, and we want to choose 3 of them. The number of ways to choose 3 letters from 7 is given by the combination formula C(n, r) = n! / (r!(n-r)!), where 'n' is the total number of items, 'r' is the number of items to choose, and '!' represents the factorial operation. Therefore, the number of ways to choose 3 letters from "article" is C(7, 3) = 7! / (3!4!) = 35.
Similarly, we must choose 3 of the 4 letters from the word "show." Following the same logic, the number of ways to choose 3 letters from "show" is C(4, 3) = 4! / (3!1!) = 4.
Next, for each of these combinations, there are 6! (6 factorial) permutations. Since we have 35 combinations from "article" and 4 combinations from "show," the total number of different arrangements is 35 * 4 * 6!.
Calculating these values, we have:
- 35 combinations from the word "article"
- 4 combinations from the word "show"
- 6! permutations of the 6 chosen letters
Therefore, the total number of different arrangements possible is 35 * 4 * 720 = 100800.