Problem Suppose we turn over cards simultaneously from two well shuffled decks of ordinary playing cards. We say we obtain an exact match on a particular turn if the same card appears from each deck; for example, the queen of spades against the queen of spades. Let $p_M$ equal the probability of at least one exact match.
Show that $$p_M=1-\frac{1}{2!}+\frac{1}{3!}-\frac{1}{4!}+...-\frac{1}{52!}$$ Hint: Let $C_i$ denote the event of an exact match on the $i^{th}$ turn. Then $p_M = P(C_1 \cup C_2 \ldots C_{52})$ Now use the the general inclusion-exclusion formula. Note that $P(C_i) = \frac1{52}$. And, hence $p_1=52(1/52)$.
Show that $p_m$ is approximately equal to $1-e^{-1}=0.632$
I do not understand why $p_1=52(1/52)$, could someone explain to me please? Thank you very much.