Evaluate.
$\frac{5!2!}{6!4!}$
Simplify your answer as much as possible.
Answer (2)
Expand the factorials: 5 ! = 120 , 2 ! = 2 , 6 ! = 720 , 4 ! = 24 .
Substitute the values into the expression: 6 ! 4 ! 5 ! 2 ! = 720 × 24 120 × 2 .
Simplify the expression by canceling out common factors: 720 × 24 120 × 2 = 6 1 × 12 1 .
Calculate the final result: 6 1 × 12 1 = 72 1 .
Explanation
Understanding the problem We are asked to evaluate the expression 6 ! 4 ! 5 ! 2 ! and simplify the result as much as possible.
Understanding factorials Let's recall the definition of the factorial function. For a positive integer n , the factorial n ! is defined as the product of all positive integers less than or equal to n . That is, n ! = n × ( n − 1 ) × ( n − 2 ) × ⋯ × 2 × 1 .
Expanding the factorials Now, let's expand the factorials in the numerator and the denominator: 5 ! = 5 × 4 × 3 × 2 × 1 = 120 2 ! = 2 × 1 = 2 6 ! = 6 × 5 × 4 × 3 × 2 × 1 = 720 4 ! = 4 × 3 × 2 × 1 = 24 So, we have 6 ! 4 ! 5 ! 2 ! = 720 × 24 120 × 2 = 17280 240
Simplifying the expression We can simplify the expression by canceling out common factors. Notice that 6 ! = 6 × 5 ! and 4 ! = 4 × 3 × 2 × 1 = 24 . Thus, we can rewrite the expression as: 6 ! 4 ! 5 ! 2 ! = 6 × 5 ! × 4 ! 5 ! × 2 ! = 6 × 4 ! 2 ! = 6 × ( 4 × 3 × 2 × 1 ) 2 × 1 = 6 × 24 2 = 144 2 = 72 1
Alternative simplification Alternatively, we can write 6 ! 4 ! 5 ! 2 ! = 6 ! 5 ! × 4 ! 2 ! = 6 × 5 ! 5 ! × 4 × 3 × 2 × 1 2 × 1 = 6 1 × 24 2 = 6 1 × 12 1 = 72 1
Final Answer Therefore, the simplified expression is 72 1 .
Examples
In probability, when calculating the likelihood of specific sequences of events, factorials are often used to determine the total number of possible outcomes. For instance, if you're arranging books on a shelf or determining the order of runners in a race, understanding how to simplify factorial expressions can help you quickly assess the probabilities involved. Simplifying expressions like this allows for easier calculations and a better understanding of the likelihood of certain events occurring in sequence.
The expression 6 ! 4 ! 5 ! 2 ! simplifies to 72 1 through the application of factorial definitions and simplification techniques. By expanding the factorials and substituting their values, we can easily reduce the expression. The final result is simplified to give 72 1 .
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