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If plq E 2P 2, find dplda: d ~Ap + 30 da...

Question

If plq E 2P 2, find dplda: d ~Ap + 30 da

If plq E 2P 2, find dplda: d ~Ap + 30 da



Answers

Solve for the variable. a) $_{n} P_{2}=30$ b) $_{n} P_{3}=990$ c) $_{6} P_{r}=30$ d) $2\left(_{n} P_{2}\right)=60$

In this problem we are given a couple of equations and we need to solve For the wearable in each of the equations. So our first equation is and Premier two equals 30. Now we can use the formula for permutations which is and for a mute R equals and factorial divided by and minus R. Factorial. Now we can use this formula to write and promote to so and Premier to is just in factorial divided by and -2 factorial And that equals 30. Now n factorial is just and times and minus one. Okay times and minus do Times and -3 and so on now and -2 times and -3 times and -4 and so on is just equal to and -2 factorial. And that is being divided by end -2 factorial as well. So we can just reduce the into our fraction and what we get is And in two and -1 equals 30. Mhm And that gives n squared minus n minus 30 equals zero. And we can break the middle term and solve the quadratic. So this will give n squared minus six. N plus five. In -30 equals zero. And we'll factor and from the first two terms n times N -6 and will factor a five from the next two terms. Next we can factor in end minus six from the left hand side. So we get and minus six plus and minus five equals zero. And now we can use the zero product property and we get n minus six equals zero, which is an equal six And also in equals -5. But we can go ahead and discard this solution because the permute operators only valid for positive numbers. So our answer is an equal six, Sir. Next problem is in Bermuda three equals 990. So we'll do the exact same thing we did before and Premier three is just and factorial divided by And -3 factorial. That equals 990. And we get and dimes and -1 Times N -3 Factorial. Using the same logic we did in the previous part. Sorry, I missed out. End -2 here. So there needs to be an end -2 over here as well and minus two Times and -3 factorial and -3 factorial and that equals Student 990. So this is divided by N -3 factorial. Just get my mind to write that down And we can reduce the entire fraction by end -3. So that gives us and times crazy. So that gives us and times and minus one Times and -2 Equals 990. And let me just move the next part dance coming in, aren't we? So that gives us yeah. Yeah. And squared minus n Times N -2 equals 990. And that gives us And Square Times and -2 minus 10 times And -2 equals 990. From there we can do the ultra pro and get and squared. Rather N cubed minus 21 squared minus N squared. Okay. Plus two n equals 990. And from there we get mm and cubed minus mhm Three M squared less to win -990 equals zero. Now from here we can basically use trial and error as we get solutions. So what we'll do is multiply the first term and the second term. So we get 990 so their magnitudes and we'll factor the expression so we get nine into Yeah, 110 And 110. Further fact arises into 11 and 10 and 10 fact rises into two and 5. So any of these as possible. So let's use trial and error. So if we evaluate the entire left hand side for so let's call the left hand side ff in. And if he find f. of nine, yeah The f. of nine equals So f. of nine equals -46, which is definitely not equal to two. And next we can try f. 11 and that we will find equals to zero. Well, let me get rid of the other questions now. They're just coming in the way. Uh huh. Yeah. So from here mhm. We have one factor so we can factories this equation as X -11 times he X squared plus bx. Let's see Equals zero. And from here we can actually do some multiplication and that gives us he x cubed plus B X squared plus E x minus 11 A x squared -11 BX. Let's see equal true. And from here we can just compare the coefficient so We know that equals to one and we know that B minus 11 equals -3. So that gives us be as being equal to eight and we know that C. Is equal 2 990 Rather -990. Okay. Okay So my bag it was 11 a. over here and that gives us the wrong value of B. She still gives us the right value of B. So that is 11 A minus three and we know A. Is just one. So that gives B equals 11 minus three Which is equal to eight. And sees in fact so this is not just see this is negative 11 C. Yeah. Yeah. Very high A- -11 C. Is equal to -990. And that gives see as equal to 90. Yeah. And from here we can determine if there are any memorial solutions by calculating the determinant B squared minus four A C. So that gives 90 squared -4 times one times eight. And this entire thing just evaluates to this entire thing evaluates to eight. Sorry, this b squared minus four A. C. And not. Yeah, this expression just do that again. So this was eight squared -4 dimes A C. So eight square to 64 And four times 90 is 360. So this gives us a negative answer. Which means there are no more real solutions. So are required answer for X Equals n equals 11 is the solution to this part. Mhm. Let me just highlight that for the next part we have six. Six pyramids are equals 30. So six permute are is just six factorial divided by what? 6- R Factorial? And that equals 30. What? Okay. And from here we can actually calculate the value of 6 -2 factorial. So 6 -1 Factorial is just equal to 30 Divided by six factorial And 30 divided by six factorial. Yeah. Yeah. The building. Sorry this is not 30 divided by six factorial. This is six factorial divided by 30. That gives us 24 and we know that 24 is just equal to four factorial. So the expressions inside the factorial must be equal which means 6 -1 equals four. And that gives R equals two As our answer to the 3rd part. Next part is quite similar but given to promote an equal 60 and that gives two factorial divided by Do -N. factorial equals 60 which gives do minus and factorial equals 60 over. Two. Factor you okay? And that gives 2- in factorial equals 30 and we know that mhm one factorial equals one. Do a factorial equals two. three factorial equals six. four factorial equals 24 And five factorial equals 120. So there is no integer solution for any factorial that equals 30. So this equation yeah is a false statement for all integers. So this has no solutions. And if you think about it it makes sense. So the only possible permutation for two or two P zero which is equal to one to be one Which is equal to two and to be too which is equal due to. So there is no possible value of N for which Do bring it an equal 60.

Okay, You know, 60 minus BC corresponds to BC, which is 52 30. So now we have 80. Do you see? It is a privilege to 1800. In other words, BC equals 22.5.

We're gonna go ahead and solve Negative. 30 d plus 12 equals 18 D. Since there's no numbers on the right side, I want to and the 30 d so that my variables are on the right side. So the 30 d will cancel. So on the left, I'm left with 12 equals and I have 18 d plus 30 d. So I have 48 de. Okay, Now I am going to get D by itself. So to do so, I'm going to divide by 48 on both sides. So now the D is all by itself. So I have d equals 12/48. Now that can be simplified, cause 12 goes into both of those numbers so I can simplify it toe 1/4. So your solution to this equation is

Here we have a ratio of permutations to calculate, and there are a number of ways we could do this problem. We could put it into the calculator real quick and be done with it. I think I'm going to take more of a fraction approach and use the permutation formulas. So four p three is four factorial, divided by four minus three factorial and four p two is poor factorial divided by four minus two factorial Okay, simplifying each of those. The top is for factorial, divided by one factorial, and the bottom is four factorial divided by two factorial. Now what we have here is a fraction divided by a fraction. We can change that to the top fraction multiplied by the reciprocal of the bottom fraction. And when we do that, we see that the four factorial can be cancelled from the top of the bottom, leaving us with just two factorial over one factorial two times one divided by one is too


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