120
This question asks for the number of distinct ways to arrange 6 people around a circular table. This is a problem related to circular permutations.
Circular Seating Arrangements Explained
When arranging items or people in a circle, the arrangement is considered the same if everyone shifts one position over. This is different from a linear arrangement where the position matters absolutely. For linear arrangements, 6 people could be seated in \(6!\) ways.
However, for a round table, we fix one person's position and arrange the remaining people relative to that person. This eliminates the duplicates caused by rotational symmetry.
Calculating Circular Permutations
The formula for the number of ways to arrange n distinct objects in a circle is given by:
Number of circular arrangements = \( (n-1)! \)
In this case, we have n = 6 people to be seated around the table.
Step-by-Step Calculation
- Identify the number of people (n): n = 6.
- Apply the circular permutation formula: \( (n-1)! \).
- Substitute n = 6 into the formula: \( (6-1)! \).
- Calculate the factorial: \( 5! \).
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
- \( 5! = 120 \).
Conclusion
Therefore, there are 120 distinct ways to seat 6 people around a round table.
Similar Questions
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Important Questions from Permutation and Combination
On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path ?
There are 6 persons arranged in a row. Another person has to shake hands with 3 of them so that he should not shake hands with two consecutive persons. In how many distinct possible combinations can the handshakes take place ?
In a tournament of Chess having 150 entrants, a player is eliminated whenever he loses a match. It is given that no match results in a tie/draw. How many matches are played in the entire tournament?
The letters A, B, C, D and E are arranged in such a way that there are exactly two letters between A and E. How many such arrangements are possible?
There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty?
