At a dance there are four married couples: Ann and Andy, Betty and Bor

Question

At a dance there are four married couples: Ann and Andy, Betty and Boris, Danielle and Dan, and Elaine and Eric. The wives select a husband at random with whom to share a dance. What is the probability that each of the four men dances with a woman other than his spouse?

A. 1/8
B. 1/6
C. 1/4
D. 1/3
E. 9/24

A. 1/8
B. 1/6
C. 1/4
D. 1/3
E. 9/24

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chetan2u · Answer

This is a derangement question, that is questions where none element is in its original position.
Direct formula for n elements =  n!(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}....+\frac{1}{n!}) 
Here n = 4 =>  4!(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}) 
 (\frac{4!}{2!}-\frac{4!}{3!}+\frac{4!}{4!})=12-4+1=9

Total ways .. First lady can choose in 4 ways, the next in 3 ways and so on => 4*3*2*1=24

P = 9/24

E

HamedATX · Answer

I think the answer is A=1/8.
The total number of ways 4 women can dance with the 4 men is 4!=24 and there is only 3 ways that none of the 4 wives won't dance with her souse. So the answer is 3/24 = 1/8

TheGraceful · Answer

There are 3 options  and 3 ways  that none of the 4 wives won't dance with her souse. So the answer is 3*3/24 = 9/24.

Stalin1994 · Answer

I thought, the first wife can choose from 3 men other than her husband - so that probability would be 3/4
Similarly, second wife can choose from 2 other men from remaining 3 men - that probability would be 2/3
After that, the third wife can choose one from the remaining 2 - that probability would be 1/2

So the each of the four men dances with a woman other than his spouse - 3/4*2/3*1/2*1/1 = 1/4 - where am I wrong?

chetan2u · Answer

The above may not be correct always..

say A1 chooses one from B2, C2 and D2.
If it is B2, then B1 also has three to choose from A2, C2 and D2.

Regor60 · Answer

No matches =

Total matches possible - (All match + 2 match + 1 match)

Total matches = 4! = 24

All match = 1 way

Two matches:

Number of ways to pick a matching pair = 4!/2!2! =

6 ways

One match:

ABCD
BCAD
CABD

Using D as the match, 2 ways.

How many ways to pick the one to match: 4

Total ways: 4*2 = 8

Total ways for no matches:

24-1-6-8 = 9

Probability: 9/24

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Oppenheimer1945 · Answer

P=D(4)/4!=9/24

D(n) is dearrangements

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At a dance there are four married couples: Ann and Andy, Betty and Bor

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At a dance there are four married couples: Ann and Andy, Betty and Bor

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