How to solve Pascal triangle method

The Pascal triangle is an arrangement of binomial coefficients in a triangular pattern. Each number in the triangle is the sum of the two numbers directly above it. The triangle starts with a single '1' at the top, and the rows are numbered starting from zero.
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Construction
The construction of Pascal's triangle is based on a simple rule:
* The rows are indexed starting from n=0 at the top.
* The entries in each row are indexed starting from k=0 from the left.
* The first and last number of each row is 1.
* The value of any other number is the sum of the two numbers directly above it.
For example, to find the number in row 3, position 1, you would add the numbers in row 2, positions 0 and 1 (1 + 2 = 3).
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Mathematical Formula
The number in the nth row and kth position (where both n and k are zero-indexed) is given by the binomial coefficient, denoted as \\binom{n}{k} or C(n,k). The formula for the binomial coefficient is:
\binom{n}{k} = \frac{n!}{k!(n-k)!}
Where n\! is the factorial of n. This formula explains why each number in the triangle represents the number of combinations of choosing k items from a set of n items.
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Properties
Pascal's triangle has many interesting properties and patterns:
* Symmetry: The triangle is symmetrical. The numbers on the left half mirror the numbers on the right half. \\binom{n}{k} = \\binom{n}{n-k}.
* Sum of Rows: The sum of the numbers in each row is a power of 2. The sum of the nth row is 2^n.
* Diagonals: The first diagonal (from the left, starting with the 1s) contains only 1s. The second diagonal contains the natural numbers (1, 2, 3, 4, ...). The third diagonal contains the triangular numbers (1, 3, 6, 10, ...).
* Hockey Stick Identity: If you start at any 1 on the edge of the triangle and sum down a diagonal of any length, the sum will be the number directly below the last number, but on the opposite diagonal. This identity is also known as the hockey stick pattern due to its shape. For example, 1+3+6+10 = 20