Evaluate.
$\frac{7!6!}{9!4!}$
Simplify your answer as much as possible.
Answer (1)
Expand the factorials: 9 ! 4 ! 7 ! 6 ! = 9 × 8 × 7 ! × 4 ! 7 ! × 6 × 5 × 4 ! .
Cancel out common terms: 9 × 8 6 × 5 .
Simplify the fraction: 3 × 3 × 2 × 4 2 × 3 × 5 = 3 × 4 5 .
The final simplified answer is: 12 5 .
Explanation
Understanding the Problem We are asked to evaluate the expression 9 ! 4 ! 7 ! 6 ! and simplify it as much as possible. Factorials are defined as n ! = n × ( n − 1 ) × ( n − 2 ) × ... × 2 × 1 .
Rewriting the Expression We can rewrite the expression by expanding the factorials to identify common terms that can be cancelled out.
Expanding Factorials 9 ! 4 ! 7 ! 6 ! = 9 × 8 × 7 ! × 4 ! 7 ! × 6 × 5 × 4 !
Cancelling Common Terms Now, we cancel out the common terms 7 ! and 4 ! from the numerator and denominator.
Simplified Expression 9 × 8 × 7 ! × 4 ! 7 ! × 6 × 5 × 4 ! = 9 × 8 6 × 5
Simplifying the Fraction Next, we simplify the fraction by canceling out common factors.
Further Simplification 9 × 8 6 × 5 = 3 × 3 × 2 × 4 2 × 3 × 5 = 3 × 4 5 = 12 5
Final Answer Therefore, the simplified value of the given expression is 12 5 .
Examples
Understanding factorials and their simplification is crucial in probability calculations, such as determining the number of ways to arrange items or the likelihood of specific events occurring in a sequence. For instance, if you're arranging books on a shelf or calculating the odds of winning a lottery, simplifying factorial expressions helps streamline the calculations and provides a clearer understanding of possible outcomes. This skill is also fundamental in more advanced mathematical fields like combinatorics and calculus, where factorial-based formulas are frequently used.
