What is the permutation of a family of 6 sitting on a round table?
Understand the Problem
The question is asking for the number of different ways a family of 6 can be arranged around a round table, which relates to the concept of permutations in combinatorics, specifically for circular arrangements.
Answer
$120$
Answer for screen readers
The number of different ways a family of 6 can be arranged around a round table is $120$.
Steps to Solve
- Understanding Circular Permutations
For circular arrangements, the formula to find the number of ways to arrange $n$ people is given by $(n-1)!$. This is because in a circle, one person's position can be fixed to avoid counting rotations as different arrangements.
- Identifying the Number of People
Here, we have a family of 6 people. Therefore, we will apply the circular permutation formula for $n = 6$.
- Applying the Formula
Using the formula for circular permutations, we can calculate the number of arrangements:
$$ \text{Number of arrangements} = (6 - 1)! = 5! $$
- Calculating Factorial
Now, we need to calculate $5!$ (which means 5 factorial):
$$ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $$
The number of different ways a family of 6 can be arranged around a round table is $120$.
More Information
In circular permutations, fixing one person helps eliminate duplicates caused by rotations. This principle is widely used in organizing events and seating arrangements.
Tips
One common mistake is using the standard factorial $n!$ instead of $(n-1)!$ for circular arrangements. To avoid this, always remember that one position in a circle can be fixed, reducing the count by one.
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