A 10 km race has 5 water stations set up around the course. What is the minimum number of people that must run in the race in order to guarantee that 10 people stop at the same set of water stations?

289

can u explain DKS,please...

The possible sets are 2^5 = 32.

We have one null-set.
We have 5 possible sets with one station stop(1 to 5).

we have 5c2 possible sets with two station stops etc ...

Now we can at max accommodate 9 runners per set.

If we add one more runner, some or other set will have 10 runners.

So (32 * 9) + 1 = 289 is ANSWER.

Why the possible sets are 2^5?

To determine the minimum number of people that must run in the race in order to guarantee that 10 people stop at the same set of water stations, we can use the Pigeonhole Principle.

The Pigeonhole Principle states that if you have n + 1 objects to place into n containers, then at least one container must have more than one object.

In this case, we can consider the 5 water stations as containers and the people running in the race as objects. We want to guarantee that there are at least 10 people who stop at the same set of water stations.

Since there are 5 water stations, we have 5 containers. In order to guarantee that 10 people stop at the same set of water stations, we need to have at least 10 + 1 = 11 people running in the race.

Therefore, the minimum number of people that must run in the race to guarantee that 10 people stop at the same set of water stations is 11.