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

Question

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) ? (A) 175 (B) 145 (C) 95 (D) 78

Tardigrade Admin · Accepted Answer

Required number of paths = ( text Number of paths upto (3,4)) ×( text Number text of paths from (3,4) text to (5,7)) = (7 !/3 ! 4 !) × (5 !/4 !)=175

Q. The number of (staircase) paths in the xy-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 , how many such paths are there if each path must pass through the point (3,4) ?

175

145

95

78

Required number of paths =( Number of paths upto (3,4))×( Number of paths from (3,4) to (5,7)) =3!4!7!​×4!5!​=175

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