Arithmetic Progression in Permutations

📌 CONCEPT: An arithmetic progression is a sequence of numbers where each term after the first is obtained by adding a fixed constant to the previous term. In this context, it will be used to determine the different sequences of ice cream flavors that the friends can get in their 7 turns.

📐 RULE / FORMULA: The formula for the nth term of an arithmetic progression is: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

💡 WORKED EXAMPLE: Let's say the 7 friends take turns to get a scoop of ice cream in a cycle of 7 turns. If A takes 3 turns to get a scoop of ice cream from the 5-person cart, then the remaining 4 friends will take 4 turns each. Using the concept of arithmetic progression, the different sequences of ice cream flavors can be determined by finding the number of permutations of 7 turns with 3 identical turns of A, which is equal to 7! / (3! * 4!). The number of different sequences of ice cream flavors is 7! / (3! * 4!) = 35.

⚠️ COMMON MISTAKE: Students may assume that the number of different sequences of ice cream flavors is equal to the number of permutations of 7 turns, which is 7!. However, this is incorrect because the 3 turns of A are identical, so we need to divide by 3! to account for the overcounting.