Find the value of [tex]\frac{6!7!-6!5!}{5!4!}[/tex]
Answer (2)
The value of the expression 5 ! 4 ! 6 ! 7 ! − 6 ! 5 ! is calculated by factoring and canceling out terms, ultimately resulting in 1230 .
;
Factor out 6 ! from the numerator: 5 ! 4 ! 6 ! 7 ! − 6 ! 5 ! = 5 ! 4 ! 6 ! ( 7 ! − 5 !) .
Rewrite 7 ! and factor out 5 ! : 5 ! 4 ! 6 ! ( 7 ! − 5 !) = 5 ! 4 ! 6 ! × 5 ! ( 42 − 1 ) .
Cancel out 5 ! and rewrite 6 ! : 4 ! 6 ! × 41 = 4 ! 6 × 5 × 4 ! × 41 .
Cancel out 4 ! and calculate the final value: 6 × 5 × 41 = 1230 . The final answer is 1230 .
Explanation
Understanding the Problem We are given the expression 5 ! 4 ! 6 ! 7 ! − 6 ! 5 ! and we need to find its value.
Factoring the Numerator First, let's factor out 6 ! from the numerator: 5 ! 4 ! 6 ! 7 ! − 6 ! 5 ! = 5 ! 4 ! 6 ! ( 7 ! − 5 !)
Rewriting Factorials Now, we can rewrite 7 ! as 7 × 6 × 5 ! and 5 ! as 5 × 4 ! . So the expression becomes: 5 ! 4 ! 6 ! ( 7 × 6 × 5 ! − 5 !) = 5 ! 4 ! 6 ! ( 42 × 5 ! − 5 !)
Factoring out 5!
We can factor out 5 ! from the parenthesis in the numerator: 5 ! 4 ! 6 ! × 5 ! ( 42 − 1 ) = 5 ! 4 ! 6 ! × 5 ! × 41
Cancelling 5!
Now we can cancel out 5 ! from the numerator and the denominator: 4 ! 6 ! × 41 We know that 6 ! = 6 × 5 × 4 ! , so we can rewrite the expression as: 4 ! 6 × 5 × 4 ! × 41
Cancelling 4!
Now, we can cancel out 4 ! from the numerator and the denominator: 6 × 5 × 41
Calculating the Value Finally, we calculate the value: 6 × 5 × 41 = 30 × 41 = 1230
Final Answer Therefore, the value of the expression is 1230.
Examples
Factorials are used in many areas of mathematics, including probability and combinatorics. For example, if you want to calculate the number of ways to arrange 6 different books on a shelf, you would calculate 6! (6 factorial), which is 6 × 5 × 4 × 3 × 2 × 1 = 720 . This concept is useful in scenarios like scheduling tasks, arranging items, and calculating probabilities in various real-world situations.
