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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 <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/mathematics/b6aa65523ae2ca11f10ced1e034c1a2b-.png /> , how many such paths are there if each path must pass through the point (3,4) ?

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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 figure
image
, how many such paths are there if each path must pass through the point $(3,4)$ ?

Permutations and Combinations

Solution:

Required number of paths
$= (\text { Number of paths upto }(3,4)) \times(\text { Number } \text { of paths from }(3,4) \text { to }(5,7)) $
$= \frac{7 !}{3 ! 4 !} \times \frac{5 !}{4 !}=175$