Navigating the world of combinatorics can be daunting. With intricate problems weaving a complex web of possibilities, finding a systematic and organized approach is often the key. Today, we introduce a method that does precisely that – we call it the *Tabulated Path Method*.

Challenge**:**
At 1:00 pm, a shopper enters Store A in a mall that houses four distinct stores. Without visiting the same store back-to-back, how many unique routes can the shopper take to return to Store A by 3:00 pm?

*Tabulated Path Method* offers a systematic way to keep track of the shopper's visits to each store at different time intervals.

Here's a brief breakdown of the table:

**At the Beginning**: The shopper starts at Store A, so there's a count of 1 in the A row under "At the beginning" and 0 for the other stores.**At 1:30pm**: The shopper can't visit the same store consecutively. Since they started at Store A, they could have moved to any of Stores B, C, or D, leading to a count of 1 for each of these stores.**At 2:00pm**: Drawing from the data at 1:30pm, the table projects where the shopper might be by 2:00pm, while adhering to the rule against consecutive visits to the same store. For instance, if the shopper finds himself in Store A by 2:00pm, his previous location at 1:30pm could have been either Store B, C, or D. By summing up the counts for these stores from the 1:30pm column, we arrive at the total of 3 for Store A in the 'At 2:00pm' column. If the shopper is positioned in Store B by 2:00pm, it signifies that he was at either Store A, C, or D at 1:30pm. To ascertain how many ways he could have reached Store B, we simply aggregate the counts for Stores A, C, and D from the 1:30pm data. This total then becomes the value for Store B in the 'At 2:00pm' column, which is 2.**At 2:30pm and At 3:00pm**: Similarly, the table calculates the possible stores at each subsequent time based on the previous time's counts.

By the time you get to 3:00pm, you can sum up the counts in the rows to determine the total number of unique routes that lead the shopper back to a specific store, in this case, Store A.

The tabulated approach is useful because it systematically counts the possible paths without repetition, ensuring that the conditions are met. However, ensuring that the calculations at each step are correct is crucial for the overall accuracy of this method.

Can you solve the challenge? The correct answer is **21**. Did you arrive at the same conclusion?

**General steps to follow in the Tabulated Path Method:**

**Initialize Your Table**: Create a table with rows for each destination (in this case, stores) and columns for each time interval or step.**Starting Point**: Mark your starting point. If our shopper starts at Store A, mark a '1' under the appropriate column and '0' for other stores.**Progressive Count**: As you move to the next column (or time interval), calculate the number of ways the shopper can move from their current location to another store, ensuring they don't revisit the same store consecutively.**Continue**: Keep populating the table until you've accounted for all time intervals.**Determine the Outcome**: Once you've filled the table, it's easy to determine the number of ways to reach a particular destination at any given time interval.

Follow up questions

If the shopper spends an additional 30 minutes shopping, bringing the total time to 5 moves, how many ways can the shopper return to Store A by 3:30 pm?

What if there were 5 stores instead of 4, but the shopper still makes only 4 moves? How many ways can he return to his starting store?

How many question can you answer correctly

2

1

0

Answer

60

52

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