An ant starts at position (0, 0) and walks north, east, south or west, one unit at a time.
How many di�erent paths of length 8 units finish at (0, 0)?
move 1 unit East: 8!/3!3!=1120
move 2 units East: 8!/2!2!2!2!=2520
move 3 units East: 8!/3!3!=1120
move 4 units East: 8!/4!4!=70
move 0 unit East: 8!/4!4!=70
Total paths: 4900
Note: if ant moves 1 unit East, must move back 1 unit West.
Take a look at the following article, and see what you can come up with:
https://betterexplained.com/articles/navigate-a-grid-using-combinations-and-permutations/
To find the number of different paths of length 8 units that finish at (0, 0), we can use the concept of combinations.
Let's visualize the problem on a coordinate plane. The ant starts at position (0, 0), and after 8 units of movement, it also ends up at (0, 0). Since the ant can only move one unit at a time in north, east, south, and west directions, there are only two possible movements for each unit:
1. Move north and then south.
2. Move east and then west.
Considering this, we can calculate the number of different paths using combinations. In this case, we need to choose 4 units to move north and 4 units to move east.
The formula for combinations is given by:
nCr = n! / (r!(n-r)!)
where n represents the total number of units and r represents the number of units to choose for a particular movement.
For our case, n = 8 and r = 4.
Plugging the values into the formula:
8C4 = 8! / (4!(8-4)!)
= 8! / (4!4!)
Now, let's calculate the value using the factorial function.
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
4! = 4 x 3 x 2 x 1 = 24
Plugging the values back into the formula:
8C4 = 40320 / (24 x 24)
= 40320 / 576
= 70
Therefore, the number of different paths of length 8 units that finish at (0, 0) is 70.
To solve this problem, we can use a simple combinatorics approach.
For each step, the ant has four possible directions to choose from: north, east, south, or west. Since the ant is starting at (0, 0) and wants to finish at (0, 0) after 8 units, the ant must take an equal number of north and south steps as well as an equal number of east and west steps.
Let's consider the number of north/south steps the ant takes. Since the ant must take an equal number of north and south steps, the number of steps in each direction can range from 0 to 4. We can represent these steps as N or S. For example, NNSS means two north steps followed by two south steps.
Similarly, the number of east/west steps the ant takes must also be equal. We can represent these steps as E or W.
Therefore, our task is to find all possible combinations of N's and S's as well as E's and W's that both add up to 8.
We can use a combinatorics formula called the binomial coefficient to find this. The formula is given as:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of steps (in this case, 8), and k is the number of steps in one direction (north/south or east/west).
Now let's calculate the number of different paths:
For the north/south steps:
C(8, 0) = 8! / (0!(8-0)!) = 1
C(8, 1) = 8! / (1!(8-1)!) = 8
C(8, 2) = 8! / (2!(8-2)!) = 28
C(8, 3) = 8! / (3!(8-3)!) = 56
C(8, 4) = 8! / (4!(8-4)!) = 70
For the east/west steps, we have the same number of combinations:
C(8, 0) = 8! / (0!(8-0)!) = 1
C(8, 1) = 8! / (1!(8-1)!) = 8
C(8, 2) = 8! / (2!(8-2)!) = 28
C(8, 3) = 8! / (3!(8-3)!) = 56
C(8, 4) = 8! / (4!(8-4)!) = 70
Finally, we can calculate the number of different paths by multiplying the number of combinations for north/south steps by the number of combinations for east/west steps:
Total number of paths = (1 * 1) + (8 * 8) + (28 * 28) + (56 * 56) + (70 * 70) = 1 + 64 + 784 + 3136 + 4900 = 7885.
Therefore, there are a total of 7885 different paths of length 8 units that finish at (0, 0).
Ah, the adventures of our intrepid ant! Let's see, the ant needs to take 8 steps to complete its journey back to the starting point. Now, every step can either be north, east, south, or west.
Think of it this way - each step is like a multiple-choice question with four options. And since the ant needs to take 8 steps in total, it's like solving an 8-question multiple-choice quiz! The number of possible paths is basically the number of ways the ant can answer these quiz questions.
So, for each step, the ant has 4 options. Since there are 8 steps in total, we need to multiply the number of options for each step, which is 4, a total of 8 times.
Mathematically, this can be expressed as 4^8, which is equal to 65,536.
Therefore, there are 65,536 different paths that our ant friend can take to end up back at our starting point, (0, 0). I hope the ant finds its way back without getting too antsy!