Question 1110201
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Interesting problem....<br>
But you didn't state the equation correctly.  The equation you show has many counterexamples.<br>
For n=6 and r=3, the formula says<br>
7P3 = 6P3+4*6P2.  But<br>
7P3 = 7*6*5 = 210
6P3 = 6*5*4 = 120
6P2 = 6*5 = 30<br>
and<br>
210 = 120 + 4*30<br>
is not true.<br>
The interesting problem is when the formula is stated correctly:<br>
(n+1)Pr = nPr + r*nP(r-1)<br>
This is easy to prove algebraically, although the nomenclature is ugly....<br>
(n+1)Pr = (n+1)(n)(n-1)(n-2)...(n-(r-2)) = (n+1)(n)(n-1)(n-2)...(n-r+2)<br>
nPr = (n)(n-1)(n-2)...(n-(r-2))(n-(r-1)) = (n)(n-1)(n-2)...(n-r+2)(n-r+1)<br>
nP(r-1) = (n)(n-1)(n-2)...(n-(r-2)) = (n)(n-1)(n-2)...(n-r+2)<br>
Then<br>
nPr + r*nP(r-1) =<br>
[(n)(n-1)(n-2)...(n-r+2)](n-r+1) + r[(n)(n-1)(n-2)...(n-r+2)] =<br>
[(n)(n-1)(n-2)...(n-r+2)][(n-r+1)+r] =<br>
[(n)(n-1)(n-2)...(n-r+2)](n+1)=<br>
(n+1)(n)(n-1)(n-2)...(n-r+2)<br>
(n+1)Pr