The given mathematical expression is:

$\frac{6!}{2!(6 - 2)!}$

This is a permutation or combination formula. Specifically, it represents a combination, calculated as:

$\binom{6}{2} = \frac{6!}{2!(6-2)!}$

Step-by-Step Solution:

1. Factorials:

   • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
   • $2! = 2 \times 1 = 2$
   • $(6-2)! = 4! = 4 \times 3 \times 2 \times 1 = 24$

2. Substitute values into the formula: $\frac{6!}{2!(6-2)!} = \frac{720}{2 \times 24}$

3. Simplify the denominator:

   • $2 \times 24 = 48$

4. Divide: $\frac{720}{48} = 15$

Final Answer:

$\frac{6!}{2!(6-2)!} = 15$

This represents the number of ways to choose 2 items from a set of 6.

Would you like further explanation or clarification?

Related Questions:

1. What does the factorial symbol (!) mean in mathematics?
2. How is this formula used in real-world applications like probability or statistics?
3. Can you demonstrate how this relates to permutations instead of combinations?
4. What happens if we calculate $\binom{6}{3}$ instead?
5. How can this formula be generalized for any $n$ and $r$?

Tip:

Always simplify factorials early to avoid unnecessary large multiplications.