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Q3:
Compute $frac{8!}{6! \times 2!}$
Solution :
Step 1: Theoretical Foundation of Factorials
The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. It is defined by the recurrence relation: $n! = n \times (n-1) \times (n-2) \times \dots \times 1$. [Per the Fundamental Definition of Factorials].
Step 2: Decomposition of the Factorial Expression
To evaluate the expression $\frac{8!}{6! \times 2!}$, we utilize the property of factorials where $n! = n \times (n-1) \times (n-2)!$. This allows for the cancellation of common terms in the numerator and denominator.
Expanding the numerator until we reach the largest factorial in the denominator:
$8! = 8 \times 7 \times 6!$
Step 3: Algebraic Substitution and Simplification
Substitute the expanded numerator back into the original expression:
$\text{Expression} = \frac{8 \times 7 \times 6!}{6! \times 2!}$
Cancel the common term $6!$ from the numerator and the denominator: [By the Property of Multiplicative Identity and Division of Like Terms]
$\text{Expression} = \frac{8 \times 7}{2!}$
Step 4: Final Calculation
Evaluate the denominator and perform the final multiplication:
- $2! = 2 \times 1 = 2$
- $\text{Expression} = \frac{56}{2}$
- $\text{Expression} = 28$
Visual Representation of Factorial Reduction
Final Solution: 28
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