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Why would a collector separate the booklet cover from the stamps
As stated previously, displaying the stamps may have been one motivation. I'd also like to add that this booklet contained 8 blocks of 6 stamps, stacked on top of each other, and if the collector is like me, he may have been nervous about leaving them stacked together in booklet form. Finally, staples from this era do not fare well after so long a time. It may be that someone didn't want them to rust and simply removed them for that reason.
I do not know if removing these stamps from the booklet affects the value (assuming you have all the contents of the booklet, still). Perhaps someone else can weigh in?