Brilli the Ant is Brilliant!

Probability Level 4

Brilli the ant sits on the Cartesian Plane at (0,0). It takes him 1 minute to crawl 1 unit. Brilli only moves horizontally and vertically. The probability that after 5 minutes he is on (2,3) is m n \frac{m}{n} where m and n are realitively prime. What is m + n 2 \frac{m+n}{2} to the nearest whole number?


The answer is 259.

Alex Segesta by Alex Segesta , 20, USA

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2 solutions

Stanley Sayson
Mar 6, 2014

Brilli has to move right 2 times, and up 3 times. He can do this in any order. We find the number of ways he can do this with 5! (total number of steps) and dividing it by repetition of a steps (2! and 3!). 5!/2!3! = 10 Now for the number of ways Brilli can move 5 times (1 for every minute) is determined through finding the number of ways brilli can move in one minute. He can move either up, down, left or right (4 number of directions). We do this five times, resulting to 4^5 = 1024. After this we divide 10 with 1024, which is equal to 5/512. 512 + 5 = 517 517/2 = 258.5 rounded up is 259

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if only it was stated that m,n are co-prime...

Roy Nadler - 7 years, 2 months ago

Well i think that it should be stated that ant can move up down right left clearly

Gauri shankar Mishra - 5 years, 4 months ago

well misunderstood the given question that he only move up or right, doesn't change the solution but pretty much the number ==

Lam Nguyen - 7 years, 2 months ago

Dang. Solved for the right answer then added m and n and just stuck in that. Silly me :(

Nicolas Bryenton - 7 years, 2 months ago

Observe that the only way for the ant to reach ( 2 , 3 ) (2,3) is to take the shortest path (ie. go right or up only). Thus, in order to reach ( 2 , 3 ) (2,3) , he has to move up by 3 and move right by 2. The number of ways to do this is thus ( 5 2 ) = 10 \left(\begin{array}{c}5\\ 2\end{array}\right)=10 .

The total number of ways to move is 4 5 4^5 , since the ant has 4 choices each minute.

Hence, the answer is 10 512 = 258.5 ≈ 259 \frac{10}{512}=258.5\approx259 .

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