Melody, i dont think he meant to hurt you :)
Jeff123- you want to find the following sum:
(n choose 0)2+(n choose 1)2+........+(n choose n)2. Now, we need to find a clearer way to express this sum. As you can see, the sum involves around binomial coefficients (no shít blarney!). This is important, because there is a special way to solve this kind of question with a method called "the combinatorical proof"-
Suppose you have a sum that contains binomial coefficients, and suppose your teacher told you to find an easier way to express that sum. One way to solve this question (the combinatorical way) is to write a "story" that matches the sum- a story that involves around you having to choose cetrain things, where the number of ways you can choose the things is the sum, and where there is an EASIER way to count the ways you can choose (a clearer formula), and by that proving that the new formula you found finds the sum.
I'll give you an example- Suppose you want to find the sum (n choose 0)+(n choose 1)+........+(n choose n) (the sum of the nth row of pascal's triangle). Now, I'll write the story that matches the sum:
In bob's class (bob is the teacher) there are n students. He needs to choose some students that will clean the class (It is possible for him to choose NO students). What is the number of ways he choose the students?-
FIRST WAY TO DESCRIBE THE NUM' OF WAYS:
bob can choose 0 students OR 1 student OR 2 students......OR n students. He has exactly (n choose k) to choose k students, Therefore he has exactly (n choose 0)+(n choose 1)+........+(n choose n) ways to choose the students.
SECOND WAY TO DESCRIBE THE NUM' OF WAYS:
bob can grab his list of his students (in the list the students are numbered from 1 to n) and think to himself:
Do i want to choose the first student?
Do i want to choose the second student?
Do i want to choose the third student?
.
.
.
Do i want to choose the nth student?
Everytime he asks himself that question, he has 2 options: the first option is to choose the student, and the second is to NOT choose him and let him leave the class. Therefore he has 2x2x2x2......x2 (n times)=2n ways to choose the students.
therefore 2n=(n choose 0)+(n choose 1)+........+(n choose n).
Can you do the same thing with the sum you gave us?
~blarney master~
Makes me smile every time i read it.
