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# Al, Bob, Cal and Don each own 1 hat. If the 4 hats are rando

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GRE Instructor
Joined: 10 Apr 2015
Posts: 1652
Followers: 57

Kudos [?]: 1577 [1] , given: 8

Al, Bob, Cal and Don each own 1 hat. If the 4 hats are rando [#permalink]  07 Nov 2018, 07:15
1
KUDOS
Expert's post
00:00

Question Stats:

35% (01:27) correct 64% (01:44) wrong based on 14 sessions
Al, Bob, Cal and Don each own 1 hat. If the 4 hats are randomly distributed so that each man receives exactly 1 hat, what is the probability that no one receives his own hat?

A) 1/8
B) 1/4
C) 1/3
D) 3/8
E) 1/2
[Reveal] Spoiler: OA

_________________

Brent Hanneson – Creator of greenlighttestprep.com

GRE Instructor
Joined: 10 Apr 2015
Posts: 1652
Followers: 57

Kudos [?]: 1577 [2] , given: 8

Re: Al, Bob, Cal and Don each own 1 hat. If the 4 hats are rando [#permalink]  08 Nov 2018, 06:01
2
KUDOS
Expert's post
GreenlightTestPrep wrote:
Al, Bob, Cal and Don each own 1 hat. If the 4 hats are randomly distributed so that each man receives exactly 1 hat, what is the probability that no one receives his own hat?

A) 1/8
B) 1/4
C) 1/3
D) 3/8
E) 1/2

I created this question to highlight many students' tendency to avoid listing and counting as a possible approach.
As you'll see, the approach is probably the fastest approach.

P(no one receives his own hat) = (# of outcomes in which no one receives his own hat)/(TOTAL number of outcomes)

# of outcomes in which no one receives his own hat
Let a, b, c and d represent the hats owned by Al (A), Bob (B), Cal (C) and Don (D)
Let's systematically list the HATS to be paired up with A, B, C, and D
A, B, C, D
b, a, d, c
b, c, d, a
b, d, a, c

c, a, d, b
c, d, a, b
c, d, b, a

d, a, b, c
d, c, a, b
d, c, b, a

So, there are 9 outcomes in which one receives his own hat

TOTAL number of outcomes
We can arrange n unique objects in n! ways
So, we can arrange the 4 hats in 4! ways (= 24 ways)
So, there are 24 possible outcomes

P(no one receives his own hat) = 9/24 = 3/8

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com

Re: Al, Bob, Cal and Don each own 1 hat. If the 4 hats are rando   [#permalink] 08 Nov 2018, 06:01
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