How Many Ordered Pairs Meet the Criteria for Ball Arrangement Probability?

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The discussion centers on calculating the number of ordered pairs (b,y) of positive integers within the range 4 ≤ b ≤ y ≤ 2007, where the probability that the balls on each end of a row of b blue balls and y yellow balls is the same color equals 1/2. The problem is framed as a conditional probability challenge, emphasizing the mathematical principles involved in combinatorial arrangements of colored objects.

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How many ordered pairs [tex](b,y)[/tex] of positive integers with [tex]4 \le b \le y \le 2007[/tex] are there such that when [tex]b[/tex] blue balls and [tex]y[/tex] yellow balls are randomnly arranged in a row, the probability that the balls on each end have the same colour is [tex]1/2[/tex]?

PS: Is there a button such that I could just click on it after I highlighted the text that I want to be included in the LaTeX tags? It's going to save me a lot of typing if there's such button (without keep typing [tex]and so on)![/tex]
 
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This just looks like some conditional probability problem. Is it homework, a challenge for the rest of us, or something else?
 

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