Pascal's triangle: and expansion of formulas and application exercise

Pascal's Triangle is a mathematical tool used to calculate binomial and combinatorial coefficients. It is named after the French mathematician Blaise Pascal.
Structure of Pascal's Triangle
The triangle is constructed as follows:

1. The first row has a 1.
2. Each subsequent row begins and ends with a 1.
3. The other numbers are calculated by adding the two numbers directly above.
Example of Pascal's Triangle
```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
```
Formula to calculate a number in Pascal's Triangle
The number in row n and column k is calculated with the formula:

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

where:

1. n is the row number
2. k is the column number
3. ! denotes the factorial (product of all positive integers up to that number)
Applications of Pascal's Triangle
1. Calculation of binomial coefficients
2. Combinatorics
3. Probability
4. Statistics
5. Algebra
6. Geometry
Properties of Pascal's Triangle
1. Symmetry
2. The sum of the numbers in each row is a power of 2
3. The sum of the numbers in each column is a power of 2
4. The triangle is used in the expansion of the binomial theorem.