5

B=ION181PC-LNN5P"2lP A-SN...

Question

B=ION181PC-LNN5P"2lP A-SN

B=ION 181P C-LNN 5P "2lP A-SN



Answers

Simplify each expression.
$$
_{5} \mathrm{P}_{2}
$$

All right, so we have a ratio of permutations to calculate and their different ways a person could do this. You could do it entirely on your calculator, One permutation at a time. I think I'm going to explore it from more of a fraction perspective. So five p three would be five factorial, divided by five minus three factorial and five p, too would be five factorial divided by five minus two factorial. I'm going to simplify each of those a little bit so the numerator would be bye factorial divided by two factorial and the denominator would be five factorial divided by three factorial. So see, we have a fraction divided by a fraction. And instead of dividing by a fraction, we could multiply by. It's reciprocal. So this is equal to the top fraction multiplied by the reciprocal of the bottom fraction notice What happens now? We can cancel the five factorial tze and what we have left is three factorial over to factorial That would be three times two times 1/2 times one, so that would just be three

To find the value of five p five. We use the permutations formula and that would give us five factorial divided by five minus five factorial. We could simplify that to five factorial divided by zero factorial and zero factorial is defined as one. So we have five factorial over one and five. Factorial is five times four times three times two times one, so that divided by one would be 1 20

To simplify this permutation, we need to find the factory or write it in a different way. We can just buy five factorial, which is the first number here over there. Difference. So it's gonna be five minus three factorial. There's gonna be five factorial over to factory. If you cancel two times one from both terms, you find that the remainder will be five times four times three since two in war are canceled. If you multiply these three numbers, you'll get 60. Because again, you five factory means five times four times three times, two times one and two. Factorial is two times one, so we can send these two terms and we get five times, four times three.

Okay to finalise common denominator of these three for actions or expressions. So for someone to start by just taking a look of the denominator and back during them. So with this, if I fact id the denominator, I'll have 12 and then peed plus five. If I factor this denominator, I'll have P Times quantity People's five. Then for this one, I'm gonna have P plus five times people's five. So basically, I'm gonna look and see what I'm missing from each one. So I know that I have to have two factors of people's five and then I know I need a P a. Many No, I need a 12. Therefore, my lease common denominator is gonna be 12 times p times People's five times People's five.


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