Consider a 25×25 grid of city streets. Let S be the points of intersection of the streets, and let P be the set of paths from the bottom left corner to the top right corner of which consist of only walking to the right and up. A point s is chosen uniformly at random from S and then a path p is chosen uniformly at random from P. Over all (s,p) pairs, the probability that the point s is contained in the path p can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?
Details and assumptions
There are 25 streets running in each direction, so S consists of 625 intersections.
where did you get this question?
It comes from an Olympiad Maths problem
Can I have a link please?
It does not have a link.
It is on paper.
Ok.
These are lame cheaters.
DO not provide them answers.
SOlved.
to ok. if you want to answer this on the site brilliant, you can by making an account but unfortunately, in every account, they have different sets of question basing on the level what you are in. this is such a very easy question.
Share your link, as I cannot create an account.
Well my answer is 49/625
To solve this problem, we need to determine the number of points in S that are contained in the paths in P.
Let's break down the problem step by step:
Step 1: Calculate the total number of points in S.
Since there are 25 streets running in each direction, the total number of intersections is 25 × 25 = 625. So, the total number of points in S is 625.
Step 2: Calculate the number of paths from the bottom left to the top right corner.
To get to the top right corner from the bottom left corner, we need to take a total of 25 steps to the right and 25 steps up. This can be thought of as arranging 25 "R" (right) and 25 "U" (up) movements in a sequence. The number of paths can be calculated using the binomial coefficient: (25 + 25) choose 25 = (50 choose 25). This equals 50! / (25! * 25!) = 196,875.
Step 3: Calculate the number of points in S that are contained in the paths in P.
To determine this, we need to count the number of paths that pass through each point in S. Since there are 625 points in S, we need to count the number of paths for each of these points and sum them up.
For any given point in S, the number of paths passing through it is equal to the number of paths from the bottom left to that point multiplied by the number of paths from that point to the top right. To calculate this, we can consider the number of steps needed to reach that point.
For example, if we consider a point in the middle row and middle column of S, it requires 12 steps to the right and 12 steps up to reach that point. We can calculate the number of paths from the bottom left to that point as (12 + 12) choose 12 = (24 choose 12).
Similarly, the number of paths from that point to the top right is the same, (24 choose 12).
Thus, the number of paths passing through that point is (24 choose 12) * (24 choose 12).
We can calculate this for each point in S and sum them up to get the total number of paths passing through the points in S.
Step 4: Calculate the probability.
To calculate the probability, we divide the total number of paths passing through points in S (found in step 3) by the total number of paths (found in step 2).
Probability = total number of paths passing through points in S / total number of paths
Step 5: Express the probability as a fraction in simplest form.
We simplify the fraction obtained in step 4 and express it as a fraction in simplest form, where a and b are coprime positive integers.
Step 6: Calculate a + b.
The value of a + b is the sum of the numerator and denominator of the simplified fraction obtained in step 5.
By following these steps, we can determine the value of a + b.