Explanation
To solve for the number of paths between two points on a grid with restricted movement (Right and Up), we use the formula for combinations of a multiset.
1. Determine the number of steps required:
To get from starting point (x1,y1)=(1,3) to ending point (x2,y2)=(5,6):
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Horizontal steps (Right, R): x2−x1=5−1=4 steps.
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Vertical steps (Up, U): y2−y1=6−3=3 steps.
2. Total number of steps:
The total number of steps (n) is the sum of horizontal and vertical steps:
3. Apply the Path Formula:
The number of ways to arrange these steps is given by the formula for combinations, where we choose the positions for either the Right steps or the Up steps:
Number of paths=(rn)=r!(n−r)!n!
Where n is the total steps and r is the number of steps in one direction (e.g., Upward steps).
4. Calculation:
Total paths=(37)=3!(7−3)!7!
Total paths=(3×2×1)×4!7×6×5×4!
Final Answer:
The number of different paths is 35. The correct option is (a).
