Like when you expand (x+y)²= (1,1)x² + (2,1)xy + (1,1)y². Its obvious in this case but when the calculation gets complicated its hard to fit in your mind
In: 0
Imagine you have a magic box with two switches on it, one labeled “x” and the other “y”. When you flip the switches, something amazing happens: the box gives you either “x” or “y”. Now, let’s see what happens when we use this magic box twice.
First, you flip the “x” switch, and the box gives you “x”. Then, you flip the “y” switch, and the box gives you “y”. So, the box gave you “xy”.
Now, let’s reverse the order. First, you flip the “y” switch, and the box gives you “y”. Then, you flip the “x” switch, and the box gives you “x”. So, the box gave you “yx”.
But wait! In math, the order doesn’t matter when we’re just adding things up. “xy” and “yx” are like two friends who like to stand together, so we can say they’re the same thing. That’s why we call them “xy” and don’t worry about the order.
Now, let’s put all this together. When you multiply (x + y) by itself, you’re using the magic box twice. So, you can get “xy” or “yx” as we talked about earlier. And that’s why you have the “xy” term in the binomial expansion.
The (1,1) thing you mentioned is like the number of ways you can choose “x” and “y” from the magic box. The numbers in the parentheses show how many times each switch was flipped. For example, (2,1) means you flipped the “x” switch twice and the “y” switch once, which gives you “xxy”. It’s like saying you got “x” twice and “y” once from the magic box.
So, when things get more complicated, just think of the magic box with its switches, and remember that “xy” and “yx” are like buddies who are the same. That should help you understand how combination works in the binomial expansion!
Latest Answers