Question: Nine family members: 5 grandchildren (3 brothers and 2 sisters) and their 4 grandparents are to be seated around a circular table. How many different seating arrangements are possible so that 2 sisters are seated immediately between some pair of brothers?

1. 120
2. 480
3. 1440
4. 2880
5. 8640

Answer:
Solution with Explanation:
Approach Solution (1):

Consider two brothers and two sisters between them as one unit:
So, now we have 6 units: {G}, {G}, {G}, {G}, {B}, and {BSSB}
These 6 unit can be arranged around a circular table in (6 – 1)! = 5! Ways
Next, analyze {BSSB} unit:

We can choose 2 brothers out of 3 for the unit in \(C^2_3\) = 3 ways;
These brothers, within the unit, can be arranged in 2! Ways: \(\{B_1,S,S,B_2\} or \{B_2,S,S,B_1\}\)
The sisters, within the unit, also can be arranged in 2! Ways: \(\{B,S_1,S_2,B\} or \{B,S_2,S_1,B\}\)
Therefore, the final answer is 5! * 3 * 2 * 2 = 1440

Correct Option: C

Approach Solution (2):

Our answer is ways in which 2 sisters (S1, S2) can sit between 2 brothers (B1, B2) + arrangement of 1 brother (B3) and 4 grandparents (G1, G2, G3, G4)
Consider an empty circular table and sister S1 comes in-
She can sit in (1 way)
Next, the second sister S2 has 2 options one to the left or one to the right of sister S1 (2 ways)
Next, we can select 2 brothers among three in 3C2 ways i.e. 3 ways who are B1 and B2. B1 can sit in 2 ways (either to the left or to the right of two sisters) and B2 can sit in 1 way.
Now there are 5 people remaining B3, G1, G2, G3, G4 and 5 positions so they can sit in 5! Ways
Final answer: 1 * 2 * 3 * 2 * 1 * 5! = 1440

Correct Option: C

Approach Solution (3):

The given condition seating arrangements are possible so that 2 sisters are seated between any two of the three brothers
2 sisters + 2 brothers; 1 brother; 4 GP
Total set; 6; which can be arranged in 5! Ways
And 2 sisters + 2 brother can be arranged; (B, S, S, B); 2! * 2! Ways and we can choose 2 out of 3 brothers in 3C2 ways
5! * 2C1 * 2C1 * 3C2; 1440

Correct Option: C

“Nine family members: 5 grandchildren (3 brothers and 2 sisters) and their 4 grandparents are to be seated around a circular table. How many different seating arrangements are possible so that 2 sisters are seated immediately between some pair of brothers?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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