<Dd> p _̄ (n) = ∏ k = 1 n − 1 (1 − k 365) <∏ k = 1 n − 1 (e − k 365) = e − n (n − 1) 730 . (\ displaystyle (\ bar (p)) (n) = \ prod _ (k = 1) ^ (n - 1) \ left (1 - (\ frac (k) (365)) \ right) <\ prod _ (k = 1) ^ (n - 1) \ left (e ^ (- (\ frac (k) (365))) \ right) = e ^ (- (\ frac (n (n - 1)) (730))).) </Dd> <P> Therefore, the expression above is not only an approximation, but also an upper bound of p (n). The inequality </P> <Dl> <Dd> e − n (n − 1) 730 <1 2 (\ displaystyle e ^ (- (\ frac (n (n - 1)) (730))) <(\ frac (1) (2))) </Dd> </Dl> <Dd> e − n (n − 1) 730 <1 2 (\ displaystyle e ^ (- (\ frac (n (n - 1)) (730))) <(\ frac (1) (2))) </Dd>

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