Evaluate the expression. 12 4 ! \frac{12}{4!} 4 ! 12

this problem, we're asked to evaluate the expression and the expression we're given is 12 over four factorial. Now there are a couple of different ways that you could choose to do this. For example, if you already know what four factorial is off the top of your head or if you have a calculator that you know how to type factorials into, but I'm gonna go ahead and expand this fully by hand so that you can see what I'm doing step by step. So I'm gonna keep that numerator the same just as 12 because there's nothing to expand there, but then in the denominator I'm going to expand four factorial out into four times three times two times one. Now I want to go ahead and fully multiply this and then of course my numerator stays the same as 12 and then if I multiply this out four times three times two times one this is going to give me 24. Now that I'm here and I don't have any factorials left in this expression I can just simplify it. So I know that 12 over left in this expression, I can just simplify it. So I know that 12 over 24, 12 is one half of 24, so this is actually one half as my fraction. Now you can write this as a decimal as well if you want to. This is also 0.5, but this is our final answer that 12 over four factorial is equal to one half. Thanks for watching and I'll see you in the next one.

14. Sequences & Series

Review of Factorials

Calculus

14. Sequences & Series

Review of Factorials

Struggling with Calculus?

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Multiple Choice

Evaluate the expression.
$\frac{12}{4!}$

$\frac14$

$\frac12$

$2$

$3$

Verified step by step guidance

Step 1: Carefully analyze the given expression. It appears to involve factorials (!), fractions, and simplifications. Factorials are defined as the product of all positive integers up to a given number, e.g., 4! = 4 × 3 × 2 × 1 = 24.

Step 2: Simplify the factorial terms in the numerator and denominator. For example, if you see 4! in both the numerator and denominator, they can cancel out, as long as they are multiplied or divided directly.

Step 3: Simplify the fractions by dividing the numerator by the denominator where possible. For instance, if you have $ \frac{12}{4!} $, calculate $ 4! $ first, then divide 12 by the result.

Step 4: Combine all simplified terms and check for any remaining operations, such as addition, subtraction, or further simplifications.

Step 5: Verify the final simplified expression matches one of the provided answer choices. Ensure all calculations are consistent and correct.