SOLUTION: 7P(n,3) = 6P(n+1,3) Not sure why the answer is n=20.

Algebra ->  Permutations -> SOLUTION: 7P(n,3) = 6P(n+1,3) Not sure why the answer is n=20.      Log On


   

Question 117274: 7P(n,3) = 6P(n+1,3)
Not sure why the answer is n=20.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!



7P%28n%2C3%29 = 6P%28n%2B1%2C3%29

Not sure why the answer is n=20.

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Use the identity P%28k%2Cr%29 = k%21%2F%28%28k-r%29%21%29
to rewrite the permutations on both sides:

7%28n%21%2F%28n-3%29%21%29++ = +6%28%28n%2B1%29%21%2F%28n%2B1-3%29%21%29

%287n%21%29%2F%28n-3%29%21++ = +%286%28n%2B1%29%21%29%2F%28n-2%29%21%29

Multiply both sides by %28n-2%29%21%2F1

%28n-2%29%21%2F1%287n%21%29%2F%28n-3%29%21++ = %28n-2%29%21%2F1+%286%28n%2B1%29%21%29%2F%28n-2%29%21%29

On the left, write %28n-2%29%21 as %28n-2%29%28n-3%29%21

%28%28n-2%29%28n-3%29%21%29%2F1%287n%21%29%2F%28n-3%29%21++ = %28n-2%29%21%2F1+%286%28n%2B1%29%21%29%2F%28n-2%29%21%29

Cancel the %28n-3%29%21's on the left and the
%28n-2%29%21's on the right:


%28n-2%29+++cross%28%28n-3%29%21%29++++%29%2F1%287n%21%29%2F%28cross%28++%28n-3%29%21%29%29++ = cross%28%28n-2%29%21%29%29%2F1+%286%28n%2B1%29%21%29%2Fcross%28%28n-2%29%21%29%29%29

7%28n-2%29n%21 = 6%28n%2B1%29%21

On the right, write %28n%2B1%29%21 as %28n%2B1%29n%21

7%28n-2%29n%21 = 6%28n%2B1%29n%21

Divide both sides by n%21

7%28n-2%29 = 6%28n%2B1%29

7n-14%29 = 6n%2B6%29

n = 20

Edwin