Question 976741
1) if nP5=6720 find n
<pre>
{{{nP5}}}{{{""=""}}}{{{n(n-1)(n-2)(n-3)(n-4)}}}{{{""=""}}}{{{6720}}}

{{{n(n-1)(n-2)(n-3)(n-4)}}}{{{""=""}}}{{{6720}}}

Those 5 factors on the left average to the middle one n-2, 
so the left side isn't far from {{{n-2)^5}}}

So the solution to

{{{(n-2)^5}}}{{{""=""}}}{{{6720}}} 

should be close to the correct answer,

{{{n-2}}}{{{""=""}}}{{{root(5,6720)}}}

{{{n-2}}}{{{""=""}}}{{{5.827386917}}}

{{{n}}}{{{""=""}}}{{{7.827386917}}}

So we try 8

{{{n(n-1)(n-2)(n-3)(n-4)}}}{{{""=""}}}{{{6720}}}
{{{8(8-1)(8-2)(8-3)(8-4)}}}{{{""=""}}}{{{6720}}}
{{{8(7)(6)(5)(4)}}}{{{""=""}}}{{{6720}}}
{{{6720}}}{{{""=""}}}{{{6720}}}

n = 8
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</pre>
2) solve equation for n 
 nP3=210
<pre>
{{{nP3}}}{{{""=""}}}{{{n(n-1)(n-2)}}}

{{{n(n-1)(n-2)}}}{{{""=""}}}{{{210}}}

These 3 factors on the left average to the middle one n-1, 
so the left side isn't far from {{{n-1)^3}}}

So the solution to

{{{(n-1)^3}}}{{{""=""}}}{{{210}}} 

should be close to the correct answer,

{{{n-1}}}{{{""=""}}}{{{root(3,210)}}}

{{{n-1}}}{{{""=""}}}{{{5.943921953}}}

{{{n}}}{{{""=""}}}{{{6.943921953}}}

So we try 7

{{{n(n-1)(n-2)}}}{{{""=""}}}{{{210}}}
{{{7(7-1)(7-2)}}}{{{""=""}}}{{{210}}}
{{{(7)(6)(5)}}}{{{""=""}}}{{{210}}}
{{{210}}}{{{""=""}}}{{{210}}}

n = 7

Edwin</pre>