Eight guests have to be seated 4 on each side of a long recangukar table. 2 particular guests desire to sit on one side of the table and 3 on the other side. the number of ways in which the sitting arrangements can be made is (a)1732 (b)1728 (c)1730 (d)1278?

Question

Answer

Assume one side as A and another as B,now having seated 3 men on side A and another 2 on B side ,we are left with 3 men,out of these 3 for side A we can choose him in 3C1 ways ,now we have two men,for B side we can choise them in 2C2 ways,now arrange all of them in 4!×4!ways,so that we have,

3C1×2C2× 4!×4!=1728

Answer

Total 8 guests divide into 2 groups 4 each
8!/4! 4! = 70
condition that 2 on one side and 3 on other
so 4C2×4C3 = 24
70×24 = 1680
on A group internal changes can be done so 2! × 2! same on b 2! ×3!( since 3 set of guest can inter change and sit)
so 2!×2!×2!×3!=48
1680+48= 1728
option (b) is correct

Answer

2!�2!�2!�3!

Answer

Solution:

Given that there are 8 guests who have to be seated 4 on each side of a long rectangular table. Let us assume that the table is as shown below.

|--------|--------|
| | |
| | |
|--------|--------|

We need to arrange the guests such that 2 particular guests sit on one side of the table and 3 on the other side. Let us assume that the two particular guests are A and B, and the remaining 6 guests are C, D, E, F, G, and H.

Case 1: A and B sit on the left side of the table

In this case, we need to arrange 2 guests out of C, D, E, F, G, and H on the left side of the table, and the remaining 2 guests along with A and B on the right side of the table.

Number of ways to select 2 guests out of 6 = 6C2 = 15

Number of ways to arrange 2 selected guests and A and B on the right side of the table = 4P2 = 12

Total number of arrangements = 15 * 12 = 180

Case 2: A and B sit on the right side of the table

In this case, we need to arrange 2 guests along with A and B on the left side of the table, and the remaining 2 guests out of C, D, E, F, G, and H on the right side of the table.

Number of ways to select 2 guests out of C, D, E, F, G, and H = 6C2 = 15

Number of ways to arrange 2 guests along with A and B on the left side of the table = 4P2 = 12

Total number of arrangements = 15 * 12 = 180

Therefore, the total number of arrangements = 180 + 180 = 360

However, the arrangement of A and B on the left or right side of the table can be interchanged, so we need to multiply the above result by 2.

Total number of arrangements = 2 * 360 = 720

But the order of arrangement of guests on each side of the table is not important. Hence, we need to divide the above result by 2.

Total number of arrangements = 720/2 = 360

Answer: (c) 1730

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