Evaluate each expression.

Question

\frac{_{4} C_{2} \cdot_{6} C_{1}}{_{18} C_{3}}

Accepted Answer

Hello. Today we are asked to evaluate the following expression. So what we are given is the combination of six and Times the combination of nine and 1 divided by the combination of 12 and four. Now in order to solve this expression, what we need to do is evaluate each of the combinations individually. And in order to do that we need to understand what the original combination states. The general combination is labeled as the combination of N and R. And the combination of N and R can be evaluated as N factorial over the quantities N minus R. Factorial times R factorial. And we're going to need to use this to evaluate the three combinations given to us. So let's go ahead and start with six. Our combination of six and four here N is going to be six and R is going to be four. So plugging this into the general combination is going to give us N factorial which is six factorial over the quantity and minus R which is six minus four factorial times R factorial which is going to be four factorial. Now let's go ahead and simplify. 6 -4 is going to give us too. So we have six factorial over two factorial times four factorial. Now remember or recall the factorial rule anytime you have a factorial, what it means is that you're taking a product of the original number times all of the numbers that are less than itself. So here we have N factorial can be evaluated as end times and minus one times N minus two and so on. So with that being said we can go ahead and extend six factorial to be six times five times four factorial and two factorial can be evaluated as two times one which is just going to be too. So what we have is two times four factorial in the numerator denominator and notice that we have a four factorial in both the numerator and denominator. We can go ahead and cancel those out and what we are left with is six times 5/2. 6 times five is going to give us 30/2 and 30/ evaluates to 15 and that's going to be the substitution for the combination of six and four. Now let's go ahead and repeat this process for the combination of nine and one, the combination of nine and one is going to give us N factorial which is nine factorial over nine and minus R. Which is nine minus one factorial times R factorial which is one factorial now one factorial is just one. So we can just go ahead and replace this with one and nine minus one is going to give us eight. So what we have is nine factorial over eight factorial times one which is just eight factorial and we can go ahead and expand nine factorial to give us nine times eight factorial in the numerator over eight factorial. And since we have an eight factorial in both the numerator and denominator we can just go ahead and cancel amount. And that's just going to leave us with nine. Now finally, let's go ahead and repeat this process. But for the combination of 12 and four, So once again evaluating this is going to give us end factorial, which in this case is going to be 12 factorial over N -R. Which is 12 -4 factorial times R factorial, which is going to be four factorial. Now 12 minus four is going to give us eight. So what we have is 12 factorial over eight factorial times four factorial. And we can go ahead and expand 12 factorial to give us 12 times times 10 times nine times eight factorial. And that's going to be over eight factorial times four factorial. Now we have an eight factorial in both the numerator and denominator. So we can go ahead and cancel this out. And expanding four factorial is going to give us four times three times two times one. So what we have is 12 times 11 times 10 times 9/4 times three times two times one. And that's going to evaluate to give us 11,880 over 24 And 11, divided by 24 is going to give us 495. And that's going to be the substitution for the combination of and four. So now that we have evaluated each of the combinations, we can go ahead and plug this back into the original expression and plugging this into the original expression is going to give us 15 Times nine over 495 now. 15 times nine over 495. Well, that's going to give us 135 over 495 and this can be simplified to be 3/11, which is going to be the answer to the given expression. And with that being said, the solution to this problem is going to be a So I hope this video helps you in understanding how to evaluate combinations. And I'll go ahead and see you all in the next video.

Evaluate each expression.
$\frac{_{4} C_{2} \cdot_{6} C_{1}}{_{18} C_{3}}$

Key Concepts

Combination Formula

Factorials

Simplifying Algebraic Fractions

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