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 4 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 of our fraction, the numerator, I have this 16 factorial. Now looking at my denominator, I have both 12 factorial and 4 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 4 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 divided by 4 fact orial which is 4 times 3 times 2 times 1. 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 4 factorial 4 times 3 times 2 times 1 gives me 24. Now dividing this in my calculator I'm going to end up getting 1,820 as my final answer. So 16 factorial divided by 12 factorial times 4 factorial is equal to 1820. Thanks for watching. And I'll see you in the next one.

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21. Combinatorics and Probability

Factorials

Precalculus

21. Combinatorics and Probability

Factorials

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Multiple Choice

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

1,820

43,680

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Verified step by step guidance

Understand that the expression $ \frac{16!}{12! \cdot 4!} $ is a combination formula, specifically $ \binom{16}{4} $, which represents the number of ways to choose 4 items from 16 without regard to order.

Recall the formula for combinations: $ \binom{n}{r} = \frac{n!}{r!(n-r)!} $. In this case, $ n = 16 $ and $ r = 4 $.

Substitute the values into the combination formula: $ \binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!} $.

Simplify the factorial expression by canceling out the common terms in the numerator and the denominator. This involves recognizing that $ 16! = 16 \times 15 \times 14 \times 13 \times 12! $, allowing $ 12! $ to cancel out.

Calculate the remaining product in the numerator: $ 16 \times 15 \times 14 \times 13 $ and divide by $ 4! = 4 \times 3 \times 2 \times 1 $ to find the number of combinations.