Question

The number of cycles of length 4 in \(S_6\) is _________.

Hint:

Each \(k\)-cycle can be written in \(k\) equivalent forms due to cyclic rotation, so divide by \(k\) after choosing \(k\) elements and permuting them.

Answer 1

Correct Answer: 90

Step 1: Formula for \(k\)-cycles.
In \(S_n\), the number of distinct \(k\)-cycles is given by \[ \frac{1}{k} \binom{n}{k} (k-1)! = \frac{n!}{k(n-k)! \, k!} = \frac{n!}{k \, (n-k)! \, k}. \] Simplifying, we use: \[ \text{Number of } k\text{-cycles in } S_n = \frac{n!}{(n-k)! \, k}. \]
Step 2: Apply for \(n=6, k=4.\)
\[ \text{Number} = \frac{6!}{(6-4)! \times 4} = \frac{720}{2! \times 4} = \frac{720}{8} = 90. \]
Step 3: Conclusion.
Hence, the number of 4-cycles in \(S_6\) is \(\boxed{90}.\)