document.write( "Question 1002501: Consider the expansion of (x+y)^n.
\n" ); document.write( "1. How many terms does the expression contain?
\n" ); document.write( "2. What is the exponent of x in the first term?
\n" ); document.write( "3. What is the exponent of y in the first term?
\n" ); document.write( "4. What is the sum of the exponents in any term of the expansion ?\r
\n" ); document.write( "\n" ); document.write( "Show work please, thanks \n" ); document.write( "

Algebra.Com's Answer #619437 by KMST(5328) $\"\"$ $\"About$

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\n" ); document.write( "is a formula/theorem that is usually proven in class, and then the proof is often forgotten.
\n" ); document.write( "1. The expression contain $\"n%2B1\"$ terms,
\n" ); document.write( "which is easy to see/count in the formula above,
\n" ); document.write( "because you see $\"1\"$ first $\"x%5En\"$ term without a $\"y\"$ , plus
\n" ); document.write( " $\"n\"$ terms showing $\"y\"$ with $\"n\"$ different numbers as exponents,
\n" ); document.write( "starting with the invisible exponent $\"1\"$ in the term $\"n%2Ax%5E%28n-1%29y=n%2Ax%5E%28n-1%29y%5E1\"$ ,
\n" ); document.write( "and going all the way to exponent $\"n\"$ in the $\"y%5En\"$ last term.
\n" ); document.write( "2. The exponent of x in the first term is $\"n\"$ because the term that we write first is $\"x%5En\"$ .
\n" ); document.write( "There is no compelling reason to write the terms in that order,
\n" ); document.write( "but it is customary, and having a certain order in mind helps keep track of all those terms.
\n" ); document.write( "3. Since the first term is $\"x%5En=x%5En%2A1=x%5En%2Ay%5E0\"$ , we can say that the exponent of y in the first term is $\"0\"$ .
\n" ); document.write( "4. The sum of the exponents in any term of the expansion is $\"n\"$ .
\n" ); document.write( "That is true for the first term, $\"x%5En=x%5En%2Ay%5E0\"$ , and is also true for all of the other terms.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "EXPLANATION OF THE FORMULA (in case you care):
\n" ); document.write( "The formula comes from the fact that
\n" ); document.write( " $\"%28x%2By%29%5En=%28x%2By%29%2A%28x%2By%29%2A%22...%22%2A%28x%2By%29%2A%28x%2By%29\"$ with $\"n\"$ factors.
\n" ); document.write( "Before simplifying, the product of those $\"n\"$ factors would have $\"2%5En\"$ products made by choosing one of the variables (x or y) from each of the $\"%28x%2By%29\"$ factors.
\n" ); document.write( "Because each of those $\"2%5En\"$ products has $\"n\"$ factors, the degree of each product (meaning the sum of the exponents of x and y) is $\"n\"$ .
\n" ); document.write( "Choosing the x from each of the $\"n\"$ $\"%28x%2By%29\"$ factors, we would get $\"x%5En\"$ . The only one way to get the product $\"x%5En\"$ is to choose the $\"x\"$ term from all of the $\"n\"$ $\"%28x%2By%29\"$ factors,
\n" ); document.write( "so you get that product only once.
\n" ); document.write( "The same can be said of $\"y%5En\"$ .
\n" ); document.write( "Because we like to put x's before y's, and the $\"%28x%2By%29%5En\"$ already had the x before the y,
\n" ); document.write( " $\"x%5En\"$ is written as the first term, and $\"y%5En\"$ is written as the last term.
\n" ); document.write( "We get other products multiple times
\n" ); document.write( "If you choose the y from one of the $\"%28x%2By%29\"$ factors and the x from the others, you get products like
\n" ); document.write( " $\"y%2Ax%2Ax%2Ax%2A%22...%22x%2Ax\"$ , $\"x%2Ay%2Ax%2Ax%2A%22...%22x%2Ax\"$ that can be written as $\"x%5E%28n-1%29%2Ay\"$ .
\n" ); document.write( "Of course there are $\"n\"$ different ways to do that, and after simplifying all those $\"n\"$ products would be accounted for in the term $\"n%2Ax%5E%28n-1%29%2Ay\"$ .
\n" ); document.write( "There are $\"n%28n-1%29%2F2\"$ ways to choose $\"2\"$ y's and $\"n-2\"$ x's,
\n" ); document.write( "and there are also $\"n%28n-1%29%2F2\"$ ways to choose $\"2\"$ x's and $\"n-2\"$ y's,
\n" ); document.write( "and that explains the coefficients in the terms $\"%28n%28n-1%29%2F2%29x%5E%28n-2%29y%5E2\"$ and $\"%28n%28n-1%29%2F2%29x%5E2y%5E%28n-2%29\"$ . \n" ); document.write( "

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