Pascal's Triangle & Combinations Exploration
To generate Pascal's triangle, you start with a row containing only the number 1. Each subsequent row is formed by adding adjacent numbers from the row above, and each row starts and ends with 1.
Here is an example of Pascal's triangle with 6 rows:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
To calculate combinations, you can use the formula: nCk = n! / (k! * (n - k)!), where n is the total number of items and k is the number of items being chosen. The exclamation mark (!) denotes the factorial of a number, which is the product of all positive integers up to that number.
For example, let's calculate "5 choose 2" (5C2):
nCk = 5! / (2! * (5 - 2)!)
Calculating the factorials:
5! = 5 * 4 * 3 * 2 * 1 = 120
2! = 2 * 1 = 2
(5 - 2)! = 3! = 3 * 2 * 1 = 6
Plugging in the values:
5C2 = 120 / (2 * 6)
= 120 / 12
= 10
So, "5 choose 2" is equal to 10.
By following this process, you can generate Pascal's triangle and calculate combinations without writing any code.