Evaluate the expression. 16 ! 12 ! ⋅ 4 ! \frac{16!}{12!\cdot4!} 12 ! ⋅ 4 ! 16 !

Here we're asked to evaluate the expression and the expression we're given is 16 factorial divided by 12 factorial times four factorial. Now looking at this, you might think that you can go ahead and cancel some stuff but remember that unless you have the same exact factorial on the top and the bottom, you can't cancel anything yet. So let's go ahead and rewrite this so that we can cancel stuff. Now remember, if you have a calculator that you know how to type factorials into, you could just type this in and get an answer. But here, I'm going to expand this out a little bit further so that you can see exactly what's happening. So on this top of our fraction, the numerator, I have this 16 factorial. Now looking at my denominator, I have both 12 factorial and four factorial. Now looking at this, our goal is going to be to cancel the highest factorial that we see in our denominator because that's going to simplify this the most. So let's go ahead and rewrite that numerator in order to cancel this 12 factorial because remember, every factorial is just a new number multiplied by the previous factorial. So this numerator can be rewritten as 16 times 15 times 14 times 13 times 12 factorial, which is exactly what we're looking to cancel. So all of this is getting divided by 12 factorial times four factorial, but now I can just go ahead and cancel this 12 factorial from both the top and the bottom. Now this leaves me in my numerator with still a pretty large number 16 times 15 times 14 times 13 and then that gets gets divided by four factorial which is four times three times two times one. Now from here we're just going to want to fully multiply out our numerator and denominator in order to get an answer. So multiplying everything out in that numerator, 16 times 15 times 14 times 13 is going to give me 43,680. Then for four factorial four times three times two times one gives me 24. Now dividing this in my calculator I'm going to end up getting eighteen twenty as my final answer. So 16 factorial divided by 12 factorial times four factorial is equal to eighteen twenty. 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

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Multiple Choice

Evaluate the expression.
$\frac{16!}{12!\cdot4!}$

1820

43,680

Verified step by step guidance

Step 1: Recognize that the given expression is $ \frac{16!}{12! \cdot 4!} $. This involves factorials, where $ n! $ (n factorial) is the product of all positive integers from 1 to $ n $.

Step 2: Simplify the numerator $ 16! $ by canceling out the $ 12! $ in the denominator. This means $ 16! = 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12! $, so the $ 12! $ terms cancel, leaving $ 16 \cdot 15 \cdot 14 \cdot 13 $.

Step 3: The expression now becomes $ \frac{16 \cdot 15 \cdot 14 \cdot 13}{4!} $. Next, calculate $ 4! $, which is $ 4 \cdot 3 \cdot 2 \cdot 1 = 24 $.

Step 4: Substitute $ 4! = 24 $ into the denominator, so the expression becomes $ \frac{16 \cdot 15 \cdot 14 \cdot 13}{24} $.

Step 5: Simplify the fraction by performing the multiplication in the numerator and then dividing by 24. This will give the final result.