Evaluate the expression. 9 ! 7 ! \frac{9!}{7!} 7 ! 9 !

Here we're asked to evaluate the expression, and the expression we're given is 9 factorial over 7 factorial. Now your first instinct might be to fully expand both the numerator and the denominator here, but but remember that we can actually write this in a way that we're going to be able to cancel some stuff and make this easier for us. So let's go ahead and do that. Now remember that any factorial is just a new number multiplied by the previous factorial. So whenever I'm writing out and expanding my numerator here, I can actually write this as 9 times 8 times 7 factorial because I see on the bottom that I have 7 factorial, so this will allow me to cancel that. So once I'm here, I can actually just fully get rid of the 7 factorial on the top and the bottom. Those are going to cancel, leaving me with just 9 times 8. So 9 times 8 is equal to 72 which gives me my final answer that 9 factorial over 7 factorial is equal to 72. 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.
$\frac{9!}{7!}$

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Verified step by step guidance

Identify the expression to evaluate: $ \frac{9!}{7!} \times \frac{7!}{2!} $.

Simplify the expression by canceling out the common terms in the numerator and the denominator. Notice that $ 7! $ appears in both the numerator and the denominator, so they cancel each other out.

After canceling, the expression simplifies to $ \frac{9 \times 8}{2!} $.

Calculate $ 2! $, which is $ 2 \times 1 = 2 $.

Divide the product of the remaining terms in the numerator by $ 2! $: $ \frac{9 \times 8}{2} $.