Bunuel wrote:

If four integers, a, b, c, and d are chosen at random, with replacement, what is the probability that ab + cd is even?

A. 1/4

B. 3/8

C. 1/2

D. 5/8

E. 3/4

We need to know the following

rules in order to solve this problem:

Even*Odd = Odd*Even = Even*Even = Even | Odd*Odd = Odd

Even + Odd = Odd + Even = Odd | Odd + Odd = Even + Even = Even---a-----b-----c-----d------ab+cd--

---E-----E-----E-----E---------E-----

---E-----E-----E-----O---------E-----

---E-----E-----O-----E---------E-----

---E-----E-----O-----O--------O-----

---E-----O-----E-----E---------E-----

---E-----O-----E-----O---------E-----

---E-----O-----O-----E---------E-----

---E-----O-----O-----O---------O-----

---O-----E-----E-----E----------E-----

---O-----E-----E-----O----------E-----

---O-----E-----O-----E----------E-----

---O-----E-----O-----O----------O-----

---O-----O-----E-----E----------O-----

---O-----O-----E-----O----------O-----

---O-----O-----O-----E----------O-----

---O-----O-----O-----O----------E-----

There are a total of 16 possibilities, of which the sum of ab and cd is even in 10 possibilities

Therefore, there are \(\frac{10}{16} = \frac{5}{8}\) possibilities(Option D) when ab+cd is even

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