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# If four integers, a, b, c, and d are chosen at random with replacement

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Math Expert
Joined: 02 Sep 2009
Posts: 44635
If four integers, a, b, c, and d are chosen at random with replacement [#permalink]

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11 Apr 2018, 21:31
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36% (00:36) correct 64% (01:59) wrong based on 23 sessions

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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
[Reveal] Spoiler: OA

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Posts: 2436
Location: India
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If four integers, a, b, c, and d are chosen at random with replacement [#permalink]

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12 Apr 2018, 00:05
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KUDOS
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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Joined: 16 Mar 2018
Posts: 36
Location: United Kingdom
GMAT 1: 650 Q45 V35
Re: If four integers, a, b, c, and d are chosen at random with replacement [#permalink]

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12 Apr 2018, 07:11
Oh wow I could not have thought of 16 possibilities like that...
My mind went like this
X=ab and Y=cd
For Sum of x and Y to be even it's possible in 2 cases
ODD +Odd is Even
Even + Even is Even
And other possibilities are
Even+Odd= Odd
Odd+ Even= Odd
So of Total 4 possibilities irrespective of what ab and cd could be the possibility fortheir summation to be even is 2/4 =1/2

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Re: If four integers, a, b, c, and d are chosen at random with replacement   [#permalink] 12 Apr 2018, 07:11
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