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When preparing for the GMAT, you might stumble upon problems that make you think outside the box—even about wires! Let’s tackle a classic example of a wire combination problem that not only challenges your math skills but also sharpens your logical reasoning. Grab your calculator; we’re about to decode how many combinations of 3 wires can be picked from 5 distinct wires, keeping the requirement that at least one of them must be a cable wire. Sounds intriguing? Let’s get to it!
First off, let’s calculate the total combinations of choosing 3 wires from 5 distinguishable wires. Calculating combinations isn’t just something you pull out of thin air—it’s methodical and precise. We can use the combination formula:
C(n, k) = n! / (k!(n-k)!)
In our case, n is 5 (the total number of wires), and k is 3 (the number of wires we want to choose). Here's how it breaks down:
C(5, 3) = 5! / (3!(5-3)!) = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10.
So, we have a lovely total of 10 combinations, which is a great start!
Now here’s the twist. We need to ensure that at least one of the wires we choose is a cable wire. Let’s think about this logically. If we have 5 wires and need to involve at least one cable wire, how many combinations would leave us without a cable wire? You might find this revelation quite surprising!
Imagine for a second that there are no cable wires in our selection. This means we’d be trying to choose 3 wires from just 2 remaining wires—yes, just two! As you can guess, you can't choose 3 wires from just 2. So, using our combo formula again:
C(2, 3) = 0.
Yup! That’s a big fat zero! There are no combinations of 3 wires without at least one cable wire.
In our case, since we began with 10 total combinations and ruled out the 0 combinations that meet our restriction—all combinations must contain at least one cable wire. Thus, we conclude confidently that we have 10 legitimate combinations!
Often, the beauty of understanding such combination problems lies beyond just the solution. They teach us essential lessons in probability and decision-making, skills that are crucial in management and in our daily lives. Every little problem you tackle builds up your capacity to face more complex challenges ahead. Ever thought about how the same logical reasoning applies in team management or project coordination? Makes you see the GMAT in a whole new light, doesn't it?
So next time you encounter a problem on the GMAT involving combinations, remember this—a firm understanding of the basics, paired with a sprinkle of logic, can help you navigate even the trickiest of scenarios. Go ahead, give yourself a pat on the back. You’ve just taken a step further in mastering these tricky yet exciting math concepts that could come up in your exam. Keep practicing, and don’t shy away from challenging problems; they’re your best teachers!