+31 question: Franklin the fly starts at the point in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After 12 steps, how many different points could Franklin end up at?
ty for giving the hint to whoever helps! And I HATE YOU to who ever will just tell me the answer. I like hints not the god darn answer. : D
+1212 Understanding the Problem
Franklin starts at (0, 0).
He can move up, down, left, or right one unit at each step.
We need to find the total number of unique points he can reach after 12 steps.
Solution Approach
This problem can be solved using a combinatorial approach.
Number of steps in each direction: Since Franklin can move in four directions, we can think of distributing 12 steps among these four directions.
Combinations: We need to find the number of ways to distribute 12 identical objects (steps) into 4 distinct boxes (directions). This is a classic stars and bars problem.
Solving the Problem
Using the stars and bars formula, the number of ways to distribute n identical objects into k distinct boxes is:
C(n + k - 1, k - 1)
In our case, n = 12 (steps) and k = 4 (directions).
So, the number of ways to distribute the steps is:
C(12 + 4 - 1, 4 - 1) = C(15, 3) = 455
Therefore, Franklin the fly can end up at 455 different points after 12 steps.
