At a Dinner Party, 5 People are to be Seated Around a Circular Table GMAT Problem Solving

Question: At a dinner party, 5 people are to be seated around a circular table. 2 seating arrangements are considered different only when the positions of the people are different relative to each other. what is the total number of different possible seating arrangements for the group?

1. 5
2. 10
3. 24
4. 32
5. 120

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement suggests that:

Given:

• 5 people are to be seated around a circular table.
• 2 seating arrangements are considered different only when the positions of the people are different relative to each other.

Find out:

• The total number of different possible seating arrangements for the group.

Here, it is given a case of circular arrangement.

The number of arrangements of n distinct objects in a row is given by n!.
The number of arrangements of n distinct objects in a circle is given by (n−1)!.

“The difference between placement in a row and that in a circle is the following:

If all objects are shifted by one position, we will obtain distinct arrangements in a row but identical relative arrangements in a circle.
Hence, for the number of circular arrangements of n objects, we have:
R= n!/n = (n−1)!”
=> (n−1)!= (5−1)! = 4! = 4*3*2*1= 24

Approach Solution 2:

The problem statement states that:

Given:

• 5 people are to be seated around a circular table.
• 2 seating arrangements are considered different only when the positions of the people are different relative to each other.

Find out:

• The total number of different possible seating arrangements for the group.

Let 1,2,3,4,5 be the people seated around a circular table.

“_ _ _ _ _” - positions

1. We can therefore fix the position of 1.

“_ _ 1 _ _”

2. We have 4*3 = 12 possible positions for left and right neighbours of 1.

“_ x 1 _ _” x e {2,3,4,5}. 4 variants
“_ x 1 y _” y e {(2,3,4,5} - {x}. 3 variants
the total number of variants is 4*3 =12

3. For each position of x1y, we have 2 possible positions for the last two people: ax1yb and bx1ya.

or
“a x 1 y _” a e {(2,3,4,5} - {x,y}. 2 variants
“a x 1 y b” b e {(2,3,4,5} - {x,y,a}. 1 variant

Therefore, N=12*2=24
Hence, the total number of different possible seating arrangements for the group = 24.

Approach Solution 3:

The problem statement informs that:

Given:

• 5 people are to be seated around a circular table.
• 2 seating arrangements are considered different only when the positions of the people are different relative to each other.

Find out:

• The total number of different possible seating arrangements for the group.

Conceptually, as we are dealing with a circular table with 5 chairs (and not a row of chairs), the table could have 5 distinct “starting chairs”. Therefore, the arrangements (going around the table) can be:

ABCDE
BCDEA
CDEAB
DEABC
EABCD

• are all identical arrangements revolving around the table.

Since we are not allowed to count each of those, we have to divide the permutation by 5.
5!/5 = 24
Hence, the total number of different possible seating arrangements for the group = 24.

“At a dinner party, 5 people are to be seated around a circular table”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The candidates can improve their skills by doing more GMAT Problem Solving questions. The GMAT Quant practice papers will help the candidates to promote their mathematical learning and to attain better marks in the exam.

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