In how many ways can 5 men and 5 women be arranged in a circle if the

Question

In how many ways can 5 men and 5 women be arranged in a circle if the men are separate and if two particular women must not be next to a particular men?

A. 432
B. 864
C. 1296
D. 1440
E. 2880

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CrackverbalGMAT · Answer

Let us look at the restrictions.

1 man cannot sit next to 2 particular woman. Let us seat him first. this is done in 1 way as all the seats are similar.

Now we have 3 women who can sit next to the man. When can seat them in 3P2 = 3!/1! = 6 ways (eg if the 3 women are A, B and C, then we can have AB, BA, AC, CA BC and CB)

Now the remaining 4 men can be seated in 4! = 24 ways and the remaining 3 women in 3! = 6 ways, as they have to be separated.

Therefore total number of ways = 1 * 6 * 24 * 6 = 864

Option B

Arun Kumar

sthahvi · Answer

why can't we do a 7! directly, and how is the factorial coming exactly ?

JoeAl · Answer

5 men can be arranged in a circle in 4!
in such cases let's apply constraints first, 2 women can arranged in 3*2 = 6 ways
and the remaining 3 women can be arranged in 3! ways

total ways : 4!*3*2*3! = 864

Choice B

ScottTargetTestPrep · Answer

Solution:

If the men must be separated and since there are an equal number of men and women, then there must be one man sitting between every two women and likewise, one woman sitting between every two men.

Recall that the number of permutations of n objects in a circular fashion is (n -1)!. Therefore, if the five men were to be seated first, there are (5 - 1)! = 4! = 24 ways. Then the five women can be seated in 5! = 120 ways. Therefore, there are a total of 24 x 120 = 2880 ways.

Now, let M be the particular man and A, B, C, D, and E be the five women. If there are no restrictions, we could have:

AMB, AMC, AMD, AME, BMA, BMC, BMD, BME, CMA, CMB,  CMD ,  CME , DMA, DMB,  DMC ,  DME , EMA, EMB,  EMC ,  EMD

We see that there are 20 seating arrangements. However, let’s say A and B are the two particular women that M can’t sit next to; then we only have 6 seating arrangements (in bold). In other words, we only have 6/20 of the 2880 sitting arrangements if a particular man can’t sit next to two particular women. Therefore, we only have 6/20 x 2880 = 864 possible seating arrangements.

Answer: B

WiziusCareers1 · Answer

To solve this circular arrangement problem, we must apply the principles of circular permutations and the "gap method" while accounting for specific constraints between individuals.

1. Arrange the Men in a Circle 
First, we place the 5 men in a circle. In circular permutations, if there are n items, they can be arranged in (n−1)! ways to account for rotational symmetry.
Ways to arrange 5 men: (5−1)! = 4! = 24 ways.

2. Identify the Gaps for Women 
To ensure that "men are separate," each woman must be placed in the gaps between the men. There are exactly 5 gaps created by 5 men in a circular arrangement.
Total gaps available: 5.

3. Handle the Constraint (Particular Women and Man) 
Let the particular man be M1 and the two particular women be W1 and W2. The condition states that W1 and W2 must not be next to M1.

M1 has two gaps adjacent to him (one to his left, one to his right).
W1 and W2 cannot take these 2 specific gaps.
To solve this, we calculate the total ways to arrange the women and subtract the "bad" cases where the constraint is violated.

Step 3a: Total ways to arrange 5 women in 5 gaps 
Once the men are fixed, the gaps are distinct positions.
Total arrangements: 5! = 120 ways.

Step 3b: Subtract cases where W1 or W2 (or both) are next to M1 
We use the Principle of Inclusion-Exclusion or direct counting for the forbidden positions:

Cases where W1 is next to M1: 
W1 can be in 2 gaps (left or right of M1).
The remaining 4 women can be arranged in the remaining 4 gaps in 4! ways.
2 × 24 = 48 ways.

Cases where W2 is next to M1: 
W2 can be in 2 gaps.
The remaining 4 women can be arranged in 4! ways.
2 × 24 = 48 ways.

Cases where BOTH W1 and W2 are next to M1: 
W1 and W2 must occupy the 2 gaps around M1. They can do this in 2! ways (W1-M1-W2 or W2-M1-W1).
The remaining 3 women can be arranged in the remaining 3 gaps in 3! ways.
2 × 6 = 12 ways.
 Total "Bad" arrangements of women: 48+48−12=84 ways.

Step 3c: Valid arrangements for women 
Valid arrangements: 120(Total) − 84(Bad) = 36 ways.

4. Final Calculation 
To find the total number of ways, multiply the arrangements of the men by the valid arrangements of the women:
Total Ways = (Arrangements of Men) × (Valid Arrangements of Women)
Total Ways = 24 × 36 = 864
 Correct Answer: B. 864

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