Evaluate each expression.

Question

\frac{_{7} P_{3}}{3!} -_{7} C_{3}

Accepted Answer

Hello. Today we are asked to evaluate the following expression. So what we are given is the permutation of eight and Over four factorial plus the combination of eight and 4. Now notice that we're given a permutation and a combination in order to evaluate these, we need to understand the general permutation and combination. The general permutation is given to us as the permutation of N and R. And this is evaluated to be N factorial over n minus R. Factorial. And for the general combination the general combination is given to us as the combination of N and R. And this evaluates to n factorial over n minus R factorial times R factorial. And for this problem understand the rule for factorial which is that if you have a number and factorial this evaluates to the product of N and all of its numbers decreasing. So n factorial evaluates to end times N -1 times N -2 and so on and so on. So we're going to be using these three rules to help us evaluate the given expression. So let's go ahead and start by evaluating the permutation of eight and 4. The permutation of eight and four is going to be N factorial which in this case is going to be eight factorial over n minus R which is eight minus four factorial Now 8 - is going to give us four factorial. So what we have is eight factorial over four factorial. And using the rule for factorial we can expand the numerator to be eight times seven times six times five times four factorial. All of that over four factorial. And since we have a four factorial in both the numerator and denominator, those are just gonna go ahead and cancel each other out. And what we are left with is eight times seven times six times five. And that's going to give us 1680. So that's the permutation evaluated. Let's go ahead and evaluate the combination. Now The combination of eight and 4 Is going to be n factorial which in this case is a factorial over N -R which is 8 -4 factorial over r factorial which is four factorial. Now eight minus four is going to give us four. So we have four factorial. We have a factorial over four factorial times four factorial. And just like the permutation, we can go ahead and expand the numerator to be eight times seven times six times five times four factorial and that's gonna be over four factorial times four factorial. Now notice that we have 24 factorial in the denominator and one in the numerator we can go ahead and cancel out one of them from both the numerator and denominator. And that leaves with was eight times seven times six times 5/4 factorial and four factorial can be rewritten As four times 3 times two times 1. Now we already know what eight times seven times six times five gives us. That's going to give us 1684 times three times two times one is going to give us 24 Now, 1680 divided by 24 is going to give us and that's going to be the combination evaluated. Now let's go ahead and take these values and plug them into our original expression. So plugging the men is going to give us over four factorial And that we're going to be adding the combination of eight and 4, which we evaluated to be 70. Now We already know what the what? 1680 over four factorial evaluates to. It's the same as this step right here So we can rewrite this entire fraction to be 70. So what we have is 70 plus 70 and that's going to give us 140 this is going to be the evaluated form of our expression. And that means that the answer to this problem is going to be C So I hope this video helps you in understanding how to evaluate permutations and combinations. And I'll go ahead and see you all in the next video

Evaluate each expression.
$\frac{_{7} P_{3}}{3!} -_{7} C_{3}$

Key Concepts

Permutation

Combination

Factorial and Simplification of Expressions

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