Knowledge Points: Principle of Least Favorable Conditions

Hello, young mathematicians! Today, we’re going to solve a fun problem about keys and safes. Imagine you’re in a company with 5 safes, each with a unique lock. Now, we have 7 employees and we want to give them keys in such a way that any 5 of them can open all the safes. Sounds tricky, right? But don’t worry, we’re going to use a cool mathematical principle to solve it. This principle is called the “Principle of Least Favorable Conditions”.

Let’s start by imagining the worst-case scenario. What if the keys to a safe are only with the two employees who are not chosen? That would be a disaster, right? Because then, the remaining 5 employees wouldn’t be able to open that safe.

So, to avoid this situation, we need to make sure that the keys to each safe are with at least three employees. This way, no matter which 5 employees are chosen, at least one of them will have the key to that safe.

Let’s visualize this with a fun analogy. Imagine the keys are like apples and the employees are like baskets. We need to distribute the apples in such a way that no matter which 5 baskets we choose, we can always find an apple in them. If we put an apple in at least three baskets, we can be sure that no matter which 5 baskets we choose, we will always find an apple!

Now, let’s do the math. We have 5 safes, and we need to distribute the keys to each safe to at least three employees. That means we need at least $5 \times 3 = 15$ keys.

And there you have it! We’ve solved the problem using the Principle of Least Favorable Conditions. Isn’t math fun? Remember, sometimes the best way to solve a problem is to think about the worst-case scenario first. Happy problem-solving, young mathematicians!