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Consider the set $S = \{2, 3, 4, \dots , 2n+1\}$ where $n$ is a positive integer larger than $2007.$ Define $\text{X}$ as the average of the odd integers in $S$ and $\text{Y}$ as the average of the even integers in $S$. What is the value of $\text{X-Y}?$

1. $0$
2. $1$
3. $n/2$
4. $n+1/2n$
5. $2008$

$Y=\frac{2+4+6+8+..+2n}{n}$

$X=\frac{3+5+7+..+(2n+1)}{n}$

$X=\frac{(2+1)+(4+1)+(6+1)+..+(2n+1)}{n}$

$X=\frac{2+4+6+8..+2n}{n}+\frac{1+1+1+1+..+n\ times}{n}$

$X=\frac{2+4+6+8..+2n}{n}+\frac{n}{n}$

$X=Y+1$

X-Y = Y+1-Y = 1

Hence,Option(2)1  is the correct choice.

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