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

Algebra.Com
Question 1201548: if P(n,2)=110 find n
Found 3 solutions by math_tutor2020, Theo, ikleyn:
Answer by math_tutor2020(3817)   (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










or

or

or
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)   (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)   (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.



RELATED QUESTIONS

find n if... (answered by math_helper)
n(n-1)=110 (answered by rapaljer)
For any integer n ≥ 3. Find n if C(n,3) + P(n,2) = 952. Please show your... (answered by ikleyn)
25. solve for n if P(n,3)=2C(n,2),n € N (answered by Edwin McCravy)
Evaluate the function. p(n)= 2^n; find... (answered by stanbon,ReadingBoosters)
Find n c. P(n, 4) = 12P(n, 2) d) C(n, 2) = 45 e) C(n, 3) = P(n, 2) f) C(n, 5) = C(n, 2). (answered by ikleyn)
If P+1 = N^2 , then find out the value of P ! Here , P= prime number & N = Normal... (answered by oscargut)
please find (little n) P (little r) if n = 6 and r =... (answered by checkley75)
Evaluate the expression. If P(n, 5) = 2520, find... (answered by stanbon)