2007 AMC 8 Problems/Problem 13

Problem

Sets $A$ and $B$ , shown in the Venn diagram, have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. Find the number of elements in $A$ .

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$\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510$

Solution

Let $x$ be the number of elements in $A$ and $B$ .

Since the union is the sum of all elements in $A$ and $B$ ,

and $A$ and $B$ have the same number of elements then,

$2x-1001 = 2007$

$2x = 3008$

$x = 1504$ .

The answer is $\boxed{C}$

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2007 AMC 8 Problems/Problem 13

Problem

Solution