Question

Trey is sewing a pentagonal button into a shirt, as shown in the figure below. The button has holes at each of its 5 corners. How many different stitches would Trey need to make in order to stitch a piece of thread across every possible pair of non-adjacent holes?

Answer 1

To stitch a piece of thread across every possible pair of non-adjacent holes in a pentagonal button with 5 corners, you can use the combination formula.

For a pentagon, there are 5 corners, and you want to choose 2 holes to stitch between, so it's a combination of 5 holes taken 2 at a time.

The combination formula is:

C(n, k) = n! / (k!(n-k)!)

Where:
- n is the total number of items to choose from (in this case, 5 holes).
- k is the number of items to choose (in this case, 2 holes).

C(5, 2) = 5! / (2!(5-2)!)

Now, calculate the combinations:

C(5, 2) = 5! / (2!(3!))

C(5, 2) = (5 * 4 * 3!) / (2! * 3!)

Now, simplify:

C(5, 2) = (5 * 4) / (2!)

C(5, 2) = (20) / (2)

C(5, 2) = 10

So, Trey would need to make 10 different stitches to stitch a piece of thread across every possible pair of non-adjacent holes in the pentagonal button.