Question

How many ways are there for a person to travel from the southwest corner to the northeast corner of an \(m \times n\) grid? Enumerate all the ways possible if the grid is \(5 \times 3\). How many ways are there if the grid is \(10 \times 10\) and no move may take the person below the main diagonal (those positions that are \(k\) steps over and \(k\) steps up from the starting point where \(1 \leq k \leq 10\) ).

Short answer

56 paths in a 5x3 grid, 16796 paths in a constrained 10x10 grid.

Step-by-step solution

Understanding the Problem

The problem asks for the number of paths from the southwest corner to the northeast corner of a grid. Moves are restricted to being right or up. Each path can be viewed as a sequence of rights (R) and ups (U).

Solution for a General m x n Grid

To move from southwest to northeast, you need to make a total of \(m\) 'R' moves and \(n\) 'U' moves. The sequence length is \(m+n\), and we need to select \(m\) positions for the 'R' moves. The number of ways to do this is given by the binomial coefficient: \(\binom{m+n}{m}\).

Solving for 5x3 Grid

In a 5x3 grid, you have to make 5 right moves and 3 up moves, total sequence length is 8. The number of sequences is given by \(\binom{8}{5}\) or \(\binom{8}{3}\) since they are equivalent. Calculate this using the formula for the binomial coefficient.

Calculate Binomial Coefficient for 5x3

Using the formula \(\binom{8}{5} = \frac{8!}{5!3!}\), calculate each factorial: \(8! = 40320\), \(5! = 120\), and \(3! = 6\). Thus, \(\binom{8}{5} = \frac{40320}{120 \times 6} = 56\).

Constraints for 10x10 Grid

Here, the person must not move below the main diagonal. This means when you have taken 'k' right steps, you can't have taken more than 'k' up steps.

Catalan Number for 10x10 Diagonal Constraint

The paths that stay below the main diagonal relate to Catalan numbers. The number of such paths in a 10x10 grid is the 10th Catalan number.

Calculate 10th Catalan Number

The n-th Catalan number is given by \(C_n = \frac{1}{n+1}\binom{2n}{n}\). For \(n=10\), calculate \(C_{10} = \frac{1}{11}\binom{20}{10}\). Compute \(\binom{20}{10} = \frac{20!}{10!10!}\) and divide by 11.

Binomial Coefficient for Catalan Calculation

Calculate \(\binom{20}{10} = 184756\) and divide by 11 to obtain \(C_{10} = 16796\), which means there are 16796 paths that do not go below the diagonal.

Key concepts

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics. It represents the number of ways to choose a set of elements from a larger set, without regard to the order. Mathematically, it is expressed as \( \binom{n}{k} \), which reads as "n choose k". Here, \( n \) is the total number of items, and \( k \) is the number of items to choose.
For example, in grid path problems, the binomial coefficient helps determine the number of paths. If you have an \( m \times n \) grid, the number of paths to go from the bottom-left to the top-right corner, moving only right and up, is \( \binom{m+n}{m} \). This is because the path consists of \( m \) right (R) moves and \( n \) up (U) moves, and you need to choose \( m \) positions for the R moves in a sequence of \( m+n \) steps.

• The value can be calculated using factorials: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• In the case of a \( 5 \times 3 \) grid, you calculate \( \binom{8}{5} \) which equals 56.

Grid Path Counting

Grid path counting is a technique used to determine the number of distinct paths on a grid where movements are restricted to specific directions. Typically, for a grid with integer dimensions \( m \times n \), the question is often to find the number of ways to move from one corner of the grid to the opposite corner.
Paths on such a grid are sequences of movements, like moving right (R) and moving up (U). For an \( m \times n \) grid, you must move \( m \) times right and \( n \) times up. The number of different paths you can take is given by the binomial coefficient \( \binom{m+n}{m} \), as earlier explained.

• Each path is a unique combination of R and U moves.
• There's a combinatorial approach to solving by arranging \( m+n \) movements as a sequence, and selecting positions for either R or U.

This calculation becomes foundational in more complex grid problems, such as those with obstacles or additional movement restrictions.

Catalan Numbers

Catalan numbers are a sequence of natural numbers that have found applications in various combinatorial mathematics problems, especially those involving recursive structures and constraints. They arise in the context of path counting problems where additional conditions are applied, such as staying within a boundary like a diagonal in a grid.
For example, in a grid where movements must stay below the main diagonal, the paths correspond to Catalan numbers. In a \( 10 \times 10 \) grid scenario, the challenge is to ensure no path crosses this main diagonal. The solution involves the 10th Catalan number.

• The formula for calculating the nth Catalan number is \( C_n = \frac{1}{n+1} \binom{2n}{n} \).
• For \( n=10 \), we compute \( C_{10} = \frac{1}{11} \binom{20}{10} \).

Catalan numbers apply to numerous problems, reflecting their versatility in combinatorial frameworks.