Vandermonde identity

So, let's say you have some numbers like 1, 2, and 3. We can call these numbers "a", "b", and "c". Now, let's say we want to find all the possible ways we can pick 2 of these numbers and multiply them together. So we want to find a x b, a x c, and b x c.

The Vandermonde identity is a way to figure out the sum of all those products we just listed. It says that if we square each number (so 1 becomes 1^2=1, 2 becomes 2^2=4, and 3 becomes 3^2=9), then the sum of all the products we listed is equal to (a+b+c)^2 - (a^2+b^2+c^2).

Now, that might sound a little confusing, so let's break it down more. First, we find all the products we can make from the three numbers: a x b, a x c, and b x c. Let's say those products are 2, 3, and 6.

Next, we square each number to get 1, 4, and 9.

Then we add up those squared numbers: 1+4+9=14.

Now, we add up our original products: 2+3+6=11.

Finally, we use the Vandermonde identity formula: (a+b+c)^2 - (a^2+b^2+c^2) = (1+2+3)^2 - (1^2+2^2+3^2) = 6^2 - 14 = 22.

So, the Vandermonde identity tells us that the sum of all the possible products we can make from three numbers is 22.

It might seem complicated at first, but the Vandermonde identity is actually pretty useful in math and science. It helps us find patterns and relationships between numbers. Plus, now you know how to wow your friends at your next math party!