Number of unique paths in Pascal's Triangle

In summary, using induction, it can be shown that the number of unique paths from entry {0,0} to entry {n,i} in Pascal's triangle is equal to pascal(n,i). The base case n=1 is easy to prove, as there is only one path to each element in the first row. It can also be noted that each value at {n,i} can be represented by nCi. Further insight can be gained by considering from which elements one can reach the ith element in the nth row with restricted movement.
  • #1
tarmon.gaidon
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Homework Statement


Let pascal(n,i) be the value of the ith element of the nth row of Pascal's triangle. Using induction show that the number of unique paths from entry {0,0} to entry {n,i} in Pascal's triangle is equal to pascal(n,i).

The Attempt at a Solution


The base case n=1 seems easy enough to prove. It is obvious that there is only one path to each of the two elements in the nth row.

It was brought to my attention that each value at {n,i} in Pascal's triangle can be represented by nCi.

That being said I am not sure how to precede. I am just looking for some insight on how to continue.

Any suggestions would be greatly appreciated.
 
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  • #2
Hint: How (from which elements) can you reach the ith element in the nth row, if movement is restricted in the appropriate way?
 

1. What is Pascal's Triangle?

Pascal's Triangle is a triangular array of numbers named after the French mathematician Blaise Pascal. The triangle is formed by starting with a row of 1 and each subsequent row is created by adding the two numbers above it together, with a 1 on each end.

2. How many unique paths are there in Pascal's Triangle?

The number of unique paths in Pascal's Triangle can be determined by the number of elements in the triangle. The first row has 1 element, the second row has 2 elements, the third row has 3 elements, and so on. Therefore, the number of unique paths in Pascal's Triangle is equal to the number of elements in the triangle.

3. How can the number of unique paths in Pascal's Triangle be calculated?

The number of unique paths in Pascal's Triangle can be calculated using the formula nCr, where n represents the row number and r represents the position of the element in that row. For example, in the 5th row of Pascal's Triangle, the 3rd element would be calculated as 5C3 = (5!)/(3!(5-3)!), which equals 10.

4. What is the significance of the number of unique paths in Pascal's Triangle?

The number of unique paths in Pascal's Triangle has many applications in mathematics, including combinatorics, probability, and geometry. It also has practical applications in fields such as computer science, where it is used in algorithms for optimizing paths and solving optimization problems.

5. Is there a limit to the number of rows in Pascal's Triangle?

There is no theoretical limit to the number of rows in Pascal's Triangle. However, as the number of rows increases, the numbers in the triangle also increase exponentially, making it difficult to accurately calculate and represent the triangle for a large number of rows.

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