Problem:
The expression
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There is exactly one term in the simplified expression for every monomial of the form , where , and are non-negative integers, is even, and . There are 1004 even values of with . For each such value, can assume any of the integer values between 0 and , inclusive, and the value of is then uniquely determined as . Thus the number of terms in the simplified expression is
This is the sum of the first 1004 odd positive integers, which is .
OR
The given expression is equal to
where the sum is taken over all non-negative integers , and with 2006. Because the number of non-negative integer solutions of is , the sum is taken over terms, but those for which and have opposite parity have a sum of zero. If is odd and is even, then is odd, so , and for some non-negative integers , and . Therefore , so . Because the last equation has non-negative integer solutions, there are terms for which is odd and is even. The number of terms for which is even and is odd is the same. Thus the number of terms in the simplified expression is
The problems on this page are the property of the MAA's American Mathematics Competitions