Consider the Cartesian (x-y) grid. If you can only move right and up, how many paths are there:
From A(0, 0) to B(6, 3)?

To solve the problem of finding the number of paths from point \( A(0, 0) \) to point \( B(6, 3) \) on a Cartesian grid, where you can only move right or up, we can use combinatorics.

### Explanation:
Each path from \( A \) to \( B \) involves exactly 6 moves to the right (R) and 3 moves up (U). Therefore, the total number of moves is \( 6 + 3 = 9 \) moves.

Out of these 9 moves, we need to choose 3 moves to be "up" moves (or equivalently, 6 moves to be "right" moves). The number of such combinations is given by the binomial coefficient \( \binom{9}{3} \).

### Calculation:
The binomial coefficient \( \binom{n}{k} \) is calculated using the formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Here, \( n = 9 \) and \( k = 3 \), so:

\[
\binom{9}{3} = \frac{9!}{3! \cdot (9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84
\]

### Conclusion:
Thus, the number of paths from \( A(0, 0) \) to \( B(6, 3) \) is **84**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How many paths are there from \( A(0, 0) \) to \( C(4, 2) \) under the same conditions?
2. What if you could move in all four directions (up, down, left, right) but still need to end at \( B(6, 3) \)?
3. How does the number of paths change if obstacles are placed on certain grid points?
4. Can you generalize the number of paths for any destination \( (m, n) \)?
5. How would you compute the number of paths if diagonal movements were allowed?

**Tip:** The binomial coefficient is a fundamental concept in combinatorics and is widely used in problems involving paths, permutations, and probability.

Learn how to calculate the number of paths from point A(0, 0) to point B(6, 3) on a Cartesian grid using combinatorics. Explore the application of binomial coefficients and solve for specific coordinates. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation