text: Finding a bound for a function involving binomial coefficients
  
  Help me please with the following question:
  
  Let $n \in N$ be fixed. I would like to find an upper and lower bound for the following function:
  
  $$f(x)={2n-x \choose n}+{x \choose 2}{2n-x \choose n-2}+\ldots+{x \choose n}-{x \choose 1}{2n-x \choose n-1}-{x \choose 3}{2n-x \choose n-3}-\ldots -{x \choose x-1}{2n-x \choose 1}.$$
  
  I know that I have to find for which $x$, $f'(x)=0$, but I don't know how to differentiate this function.
  
  Thank you.
  
  -
  Are you looking for a maximum or just an upper bound? – wj32 Nov 1 '12 at 22:19
  The way you've written $f(x)$ only makes sense for even $n$; otherwise it's not clear how the dots are to be interpreted. – joriki Nov 2 '12 at 0:16
  You're choosing $n$ items from $2n$ items, distinguished according to how many you choose from some particular subset $X$ of $x$ items, and you're considering the excess of selections with an even number of items in $X$ over selections with an odd number of items in $X$. For $x=0$ or $x=2n$, all selections have an even number of items in $X$, so the best upper bound is $\binom{2n}n$. A lower bound is more difficult to obtain; I would suspect that it's attained at $x=n$, and $\sum\binom nk\binom n{n-k}(-1)^k=\sum\binom nk^2(-1)^k$ can be positive or negative. – joriki Nov 2 '12 at 1:18
explanation: The document is a math question posted on a forum or similar platform. The user is asking for help in finding upper and lower bounds for a function involving binomial coefficients. The function is defined in a specific form, and the user mentions that they know they need to find the critical points by setting the derivative to zero but are unsure how to differentiate the function. There are also comments from other users providing additional insights and clarifications.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Finding a bound for a function involving binomial coefficients
  
  Help me please with the following question:
  
  Let $n \in N$ be fixed. I would like to find an upper and lower bound for the following function:
  
  $$f(x)={2n-x \choose n}+{x \choose 2}{2n-x \choose n-2}+\ldots+{x \choose n}-{x \choose 1}{2n-x \choose n-1}-{x \choose 3}{2n-x \choose n-3}-\ldots -{x \choose x-1}{2n-x \choose 1}.$$
  
  I know that I have to find for which $x$, $f'(x)=0$, but I don't know how to differentiate this function.
  
  Thank you.
  
  -
  Are you looking for a maximum or just an upper bound? – wj32 Nov 1 '12 at 22:19
  The way you've written $f(x)$ only makes sense for even $n$; otherwise it's not clear how the dots are to be interpreted. – joriki Nov 2 '12 at 0:16
  You're choosing $n$ items from $2n$ items, distinguished according to how many you choose from some particular subset $X$ of $x$ items, and you're considering the excess of selections with an even number of items in $X$ over selections with an odd number of items in $X$. For $x=0$ or $x=2n$, all selections have an even number of items in $X$, so the best upper bound is $\binom{2n}n$. A lower bound is more difficult to obtain; I would suspect that it's attained at $x=n$, and $\sum\binom nk\binom n{n-k}(-1)^k=\sum\binom nk^2(-1)^k$ can be positive or negative. – joriki Nov 2 '12 at 1:18
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a math question posted on a forum or similar platform. The user is asking for help in finding upper and lower bounds for a function involving binomial coefficients. The function is defined in a specific form, and the user mentions that they know they need to find the critical points by setting the derivative to zero but are unsure how to differentiate the function. There are also comments from other users providing additional insights and clarifications.
  
  Label: question