Shuvo Loves Ice Creams

Probability Level 3

My friend Shuvo loves ice creams and he studies at Khulna University. He visits the Ice Cream shop almost every day while he returns home from University. Shuvo's University is at the lowest right corner U U , His home is at the point H H and The ice cream shop is at the point I I . She is only allowed to move either upwards or to the left.

Today, as his mother is very sick, she has to go home quickly. As such, he decides to avoid temptation and skip visiting the ice cream shop completely.

How many different paths are there for her to take?


The answer is 1743.

Arindam Kumar Paul by Arindam Kumar Paul , 25, Bangladesh

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1 solution

For going home , if we consider the University as the origin, then her home is at ( 8 , 6 ) \left ( 8,6 \right ) , and the ice Cream Shop is at the point ( 5 , 4 ) \left ( 5,4 \right ) So, She can go home from the University by using 6 14 C = 3003 _{6}^{14}\textrm{C}=3003 paths.

But, the ice cream shop is at the point ( 5 , 4 ) \left ( 5,4 \right ) and she can go to the ice Cream Shop by using 4 9 C = 126 _{4}^{9}\textrm{C}=126 paths. And, she can go home from ice cream shop by using 2 5 C = 10 _{2}^{5}\textrm{C}=10 paths. So, If she skips all paths that including Ice Cream shop, She have the total 6 14 C − ( 4 9 C × 2 5 C ) = 3003 − ( 126 × 10 ) = 3003 − 1260 = 1743 _{6}^{14}\textrm{C}-\left (_{4}^{9}\textrm{C} \times _{2}^{5}\textrm{C} \right )=3003-(126\times 10)=3003-1260=1743 paths to go home .

So, The answer is 1743 .

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