SOLUTION: if P(n,2)=110 find n

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Question 1201548: if P(n,2)=110 find n
Found 3 solutions by math_tutor2020, Theo, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: n = 11


Work Shown:

P(n,r) = n*(n-1)*(n-2)*...*(n-r+1) is the permutation function

When r = 2,
P(n,2) = n*(n-1)

So,
P(n,2) = 110
n(n-1) = 110
n^2-n = 110
n^2-n-110 = 0

We could use a trial-and-error approach to factor.
But I find the most efficient pathway is to use the quadratic formula.
Think of n as x
We have x^2-x-110 = 0 in the form ax^2+bx+c = 0
a = 1
b = -1
c = -110
x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29

x+=+%28-%28-1%29+%2B-+sqrt%28%28-1%29%5E2+-+4%281%29%28-110%29%29%29%2F%282%281%29%29

x+=+%281+%2B-+sqrt%281%2B440%29%29%2F%282%281%29%29

x+=+%281+%2B-+sqrt%28441%29%29%2F%282%29

x+=+%281+%2B-+21%29%2F%282%29

x+=+%281+%2B+21%29%2F%282%29 or x+=+%281+-+21%29%2F%282%29

x+=+%2822%29%2F%282%29 or x+=+%28-20%29%2F%282%29

x+=+11 or x+=+-10
Your steps for the quadratic formula do not need to be this verbose.

The solutions x = 11 or x = -10 lead back to n = 11 or n = -10.

The permutation function only allows positive integers for n.
Therefore, we eliminate n = -10 and circle n = 11 as the final answer.

Check:
P(n,2) = n*(n-1)
P(11,2) = 11*(11-1)
P(11,2) = 11*10
P(11,2) = 110
The answer is confirmed.

Real world application:
There are 11 people running for the government positions "treasurer" and "secretary". This means order matters.
There are 11*10 = 110 different permutations possible.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
n = 11.
you get p(n,2) = n! / (n-x)! = 11! / 9! = 11 * 10 * 9! / 9! = 11 * 10 = 110.
the simple rule is:
p(n,1) = n
p(n,2) = n * (n-1)]
p(n,3) = n * (n-1) * (n-2)
p(n,4) = n * (n-1) * (n-2) * n-3)
etc.
if n is 11, then:
p(11,1) = 11
p(11,2) = 11 * 10 = 110
p(11,3) = 11 * 10 * 9 = 990
p(11,4) = 11 * 10 * 9 * 8 = 7920
etc.....
looking at p(11,4), you get:
p(11,4) = 11! / (11-4)! = 11! / 7! = 11 * 10 * 9 * 8 * 7! / 7! = 11 * 10 * 9 * 8 = 7920.

Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.
if P(n,2) = 110, find n.
~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The P(n,2) formula is

    P(n,2) = n*(n-1).


They want you find n from equation

    n*(n-1) = 110.        (1)


At this point, you can guess the solution for (1) mentally (n=11),
since you have the product of two consecutive integers in left side.


Alternatively, you can write it in the standard form quadratic equation

    n^2 - n - 110 = 0     (2)


and solve it by factoring left side

    (n-11)*(n+10) = 0,

or you can solve it using the quadratic formula.


In any way, you will obtain the same 


ANSWER.  n = 11.

Solved.