Integers \( a, b, c \) and \( d \) not necessarily distinct, are chosen independently and at ran...

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Integers \( a, b, c \) and \( d \) not necessarily distinct, are chosen independently and at random from the set \( S=\{0,1,2,3, \ldots \ldots .2006,2007\} \). If the probability that \( \mid a d \) \( -b c \| \) is even, is \( \frac{p}{q} \) where \( p \) and \( q \) are relatively prime the the value of \( (p+q) \).
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