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Counting - Permutations

SOURCE:COMPETITION
Number of Problems: 2.
FOR PRINT ::: (Book)

Problem Num : 1

Type: Application

Topic:Counting

Theme:Counting
Adjustment# :

Difficulty: 2

Category Permutations

Analysis

Solution/Hint

Problem Num : 2

Type:

Topic:Counting

Adjustment# : 0

Difficulty: 1

'
Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$ . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?
$mathrm{(A)} 240qquadmathrm{(B)} 360qquadmathrm{(C)} 480qquadmathrm{(D)} 540qquadmathrm{(E)} 720$
'

Category Permutations

Analysis

Solution/Hint
For the first man, there are $10$ possible seats. For each subsequent man, there are $4$ , $3$ , $2$ , and $1$ possible seats. After the men are seated, there are only two possible arrangements for the five women. The answer is $10cdot 4cdot 3cdot 2cdot 1cdot 2 = oxed{480}$ .