In Exercises 21–28, evaluate each expression.

Question

Accepted Answer

Hello. Today we are asked to evaluate the following expression, what we are given is three minus the permutation of five and two over the permutation of six and five. Now in order to solve this, we need to know how to evaluate the given permutations. And in order to evaluate the permutations we need to understand the general permutation. The general permutation is given as the permutation of N and R. And this can be evaluated as N factorial over n minus R factorial. So we need to go ahead and use this rule to evaluate the given permutations. Let's go ahead and start by evaluating the permutation of five and by using the general form permutation this is going to give us n factorial which in this case is going to be five factorial over n minus R which is five minus two factorial. Five minus two factorial is going to give us three factorial. So we have five factorial over three factorial. Now remember the rule for factorial if you're given a number and factorial this is the product of the number itself times all of the numbers decreasing by one. So n factorial evaluates to end times N -1 times N -2 and so on and so on. So using this rule we can expand the numerator five factorial to be five times four times three factorial and that's gonna be over three factorial. Notice how we have a three factorial in both the numerator and denominator we can just go ahead and cancel amount and that's going to leave us with five times four which is going to give us 20. So the permutation of five and two evaluates to 20. Let's go ahead and evaluate the permutation of six and 5. The permutation of six and five is going to give us n factorial which in this case is six factorial over and minus R. Which is six minus five factorial. Now here 6 -5 is just going to give us one. So we have six factorial over one factorial and one factorial just evaluates to be one. So we're just left with six factorial and expanding. This is going to give us six times five times four times three times two times one. And this is going to give us 720. So we have a substitution for the permutation of six and five. Now let's go ahead and take these values and plug them into the original expression. Doing this is going to give us 3 - over 720 and 20 over 720 can simplify to 1/36. So we have three minus 1/36. Now we can go ahead and put 3/1 and in order to subtract the fractions together we need to have the same denominator here, the lowest common denominator is 36. So if we multiply 3/1 by 36/36. This is going to give us Over 36 -1/ And 108 -1 is going to give us 107 over 36 and this is going to be the simplified form of the given expression. And that means that the answer to this problem is going to be D. So I hope this video helps you in understanding how to evaluate a permutation, and I'll go ahead and see you all in the next video.