Find #n# if #""^nP_3=20n#?

Question

The answer is 6 but I'm unsure how to get to it.

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Answer 1 · 2017-10-11T23:56:13

#n=6#

#""^nP_r# is defined as #(n!)/(r!)#.

Thus, #""^nP_3=(n!)/(3!)#. We know that this is equal to #20n#.

#(n!)/(3!)=20n#

Multiply both sides by #(3!)/n#:
#(n!)/n=20*3!#

We know that #3! =3*2*1=6#, and #n!=n*(n-1)!#.

#(n*(n-1)!)/n=20*6#
#(n-1)! =120#

Since #5! =5*4*3*2*1=120#, #n-1=5#.

The final answer is #n=5+1=6#.