SOLUTION: Consider the expansion of (x+y)^n. 1. How many terms does the expression contain? 2. What is the exponent of x in the first term? 3. What is the exponent of y in the first te

Question 1002501: Consider the expansion of (x+y)^n.
1. How many terms does the expression contain?
2. What is the exponent of x in the first term?
3. What is the exponent of y in the first term?
4. What is the sum of the exponents in any term of the expansion ?
Show work please, thanks
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

is a formula/theorem that is usually proven in class, and then the proof is often forgotten.
1. The expression contain terms,
which is easy to see/count in the formula above,
because you see first term without a , plus
terms showing with different numbers as exponents,
starting with the invisible exponent in the term ,
and going all the way to exponent in the last term.
2. The exponent of x in the first term is because the term that we write first is .
There is no compelling reason to write the terms in that order,
but it is customary, and having a certain order in mind helps keep track of all those terms.
3. Since the first term is , we can say that the exponent of y in the first term is .
4. The sum of the exponents in any term of the expansion is .
That is true for the first term, , and is also true for all of the other terms.

EXPLANATION OF THE FORMULA (in case you care):
The formula comes from the fact that
with factors.
Before simplifying, the product of those factors would have products made by choosing one of the variables (x or y) from each of the factors.
Because each of those products has factors, the degree of each product (meaning the sum of the exponents of x and y) is .
Choosing the x from each of the factors, we would get . The only one way to get the product is to choose the term from all of the factors,
so you get that product only once.
The same can be said of .
Because we like to put x's before y's, and the already had the x before the y,
is written as the first term, and is written as the last term.
We get other products multiple times
If you choose the y from one of the factors and the x from the others, you get products like
, that can be written as .
Of course there are different ways to do that, and after simplifying all those products would be accounted for in the term .
There are ways to choose y's and x's,
and there are also ways to choose x's and y's,
and that explains the coefficients in the terms and .
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