Evaluate each expression.

Question

\frac{_{7} C_{3}}{_{5} C_{4}} - \frac{98!}{96!}

Accepted Answer

Hello. Today we're going to be evaluating the following expression. So what we are given is the combination of six and two over the combination of six and -120 factorial over 118 factorial. Now since we are given combinations in order to evaluate the combinations, we need to understand the general combination. The general combination is given to us as the combination of N and R. And this can be evaluated to be N factorial over n minus R. Factorial times R. Factorial. And for any factorial given to us in this case and factorial we can expand this to be the number times itself and all of its decreasing numbers. So in this case here is going to be n times n minus one times n minus two and so on and so on. So let's go ahead and start by evaluating the combination of six and 2, the combination of six and two while six takes the place of n and two takes the place of our. So plugging this into the general combination is going to give us n factorial which in this case is six factorial over n minus R which is six minus two factorial times R factorial which is going to be two factorial Now 6 -2 is going to give us four. So what we have is six factorial over four factorial and two factorial can be rewritten as two times one. And that's just going to evaluate the two. So in the denominator we have four factorial times two. Now in the numerator we can go ahead and expand six factorial to give us six times five times four factorial and that's going to be over four factorial times two. Now notice that we have a four factorial in both the numerator and denominator. We can go ahead and cancel us out and that's going to leave us with six times 5/2. Six times five is going to give us 30. So we have 30/2 and that's going to evaluate to 15. So we have a substitution for the combination of six and two. It's going to be 15. Now let's go ahead and evaluate the combination of six and four. The combination in six and four is going to give us n factorial which is six factorial over n minus R. Which is six minus four factorial times R factorial which is going to be four factorial Now 6 -4 is going to give us too. So we have six factorial over two factorial which is just two times four factorial. Now notice that these two right ratios are exactly the same. It's just that the products are ordered differently in the denominator but we know that six factorial over four factorial times two evaluates to 15 With that being said six factorial over two times four factorial is also going to evaluate to 15. So let's go ahead and plug this back in to the original expression since both of the combinations evaluate 2, 15 we have 15/15 minus 100 and factorial over 118 factorial. Now, just like before we can go ahead and expand 100 and 20 factorial and that's going to give us 100 and 20 times Times 118 factorial over factorial. And since we have 118 factorial in both the numerator and denominator, we can just go ahead and cancel this out. And what we are left with is 120 times 119 and that's going to give us 14,280. So we have 15/ -14280. Now 15/15 can be evaluated to just one and that's going to give us one minus 14,280 which is going to give us negative 14, and that's going to be the answer to the given to the given expression and that means that the answer to this problem is going to be be 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.
$\frac{_{7} C_{3}}{_{5} C_{4}} - \frac{98!}{96!}$

Key Concepts

Combination Formula

Factorials

Simplifying Factorial Expressions

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