Apoorva2705
Bunuel
Official Solution:
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 between any two of the three brothers?
A. 120
B. 480
C. 1440
D. 2880
E. 8640
Consider two brothers and two sisters between them as one unit: {BSSB}.
So, now we have 6 units: {G}, {G}, {G}, {G}, {B}, and {BSSB}.
These 6 units 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.
Answer: C
Hi,
I have approached this problem slightly differently.
1. We chose 1 brother in 3C1 ways and place him on a circular table in 1 way
2. We chose 1 sister in 2C1 ways and place her on the table in 2 ways. (left or right of the brother)3. The second sister can sit on the table in 1 way only
4. We chose 1 brother in 2C1 ways and place him on the table in 1 way
5. Remaining 5 people can sit in 5! ways
My answer: 3*1*2*2*1*2*1*5! = 2880
VeritasKarishma Bunuel: Could you please tell me whats wrong with my method?
There is double counting in your method. Say you have S1, S2 and B1, B2, B3.
Say, for your first brother, you select B1 and make him sit anywhere. Then you chose S1 on his right and S2 on further right. Then you select B2 to sit on S2's right. And the others are placed around.
Now imagine for your first brother, you selected B2. Then you chose S2 on his left, S1 on her left and you chose B1 to sit on S1's left. Then all the rest around.
In your answer, these are two different cases but in the actual answer, this is a single arrangement only.
Simply put, say you want to arrange 5 people such that A and B sit next to each other on a circular table, you must consider them as a unit and arrange them in 2 ways.
If you instead use your method, you will select one of them in 2 ways and put the other next to him in 2 ways (left or right) doubling the arrangements.
Hence, whenever you need to make some people sit together, it is always better to use the method of tying them together as one unit and then arranging.