Multinomial theorem
The multinomial theorem is a math rule that helps us expand expressions with lots of terms all raised to different exponents. Imagine you have three numbers, let's call them a, b, and c. The multinomial theorem tells us that if we raise them to any powers, say x, y, and z, respectively, then we can write the result as a polynomial (which is just a fancy word for an expression with lots of terms) by multiplying out all the possible combinations of terms.
Let's say we want to find out what (a+b+c)^2 is. We can use the multinomial theorem to expand this expression. To make things simpler, let's use little numbers as exponents: (a+b+c)^(1+1). In other words, each base will be raised to the power of 1.
The first term in our expansion will be a^2. The second term will be 2ab, which means we have to multiply a by b twice: once when we pick it from the first set of parentheses, and once when we pick it from the second set. The third term will be 2ac, and the fourth term will be 2bc. Finally, there will be a term that's just b^2, and one that's just c^2.
So, (a+b+c)^2 = a^2 + 2ab + 2ac + 2bc + b^2 + c^2.
Now, this may seem like a lot of work, especially if we have to keep raising lots of different numbers to different powers. But the multinomial theorem simplifies things a lot by telling us exactly how many terms we need to multiply, and what the coefficients (which are the numbers in front of the a^2, ab, ac, etc.) will be.
So, if we wanted to expand (a+b+c)^3, we could use the multinomial theorem to find that the result would be:
(a+b+c)^3 = a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3.
Phew, that's a lot of terms! But thanks to the multinomial theorem, we know exactly how many there will be, and we can easily find each one by multiplying the bases in all possible combinations.
Let's say we want to find out what (a+b+c)^2 is. We can use the multinomial theorem to expand this expression. To make things simpler, let's use little numbers as exponents: (a+b+c)^(1+1). In other words, each base will be raised to the power of 1.
The first term in our expansion will be a^2. The second term will be 2ab, which means we have to multiply a by b twice: once when we pick it from the first set of parentheses, and once when we pick it from the second set. The third term will be 2ac, and the fourth term will be 2bc. Finally, there will be a term that's just b^2, and one that's just c^2.
So, (a+b+c)^2 = a^2 + 2ab + 2ac + 2bc + b^2 + c^2.
Now, this may seem like a lot of work, especially if we have to keep raising lots of different numbers to different powers. But the multinomial theorem simplifies things a lot by telling us exactly how many terms we need to multiply, and what the coefficients (which are the numbers in front of the a^2, ab, ac, etc.) will be.
So, if we wanted to expand (a+b+c)^3, we could use the multinomial theorem to find that the result would be:
(a+b+c)^3 = a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3.
Phew, that's a lot of terms! But thanks to the multinomial theorem, we know exactly how many there will be, and we can easily find each one by multiplying the bases in all possible combinations.
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