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  4. The number of (staircase) paths in the x y-plane from (0,0) to (7,5) where each such path is made up of individual steps going one unit upward (U) or one unit to the right (R). One such path is shown in figure

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Q. The number of (staircase) paths in the $x y$-plane from $(0,0)$ to $(7,5)$ where each such path is made up of individual steps going one unit upward $(U)$ or one unit to the right $(R)$. One such path is shown in figureMathematics Question Image

Permutations and Combinations

Solution:

The number of paths
$=\text { the number of ways of arranging } 5 U^{\prime} S \text { and } 7 R^{\prime} \text { s. } $
$=\frac{12 !}{5 ! 7 !}={ }^{12} C_5$