Answer:
[tex]P(n,4) = 840[/tex]
[tex]P(n,r) = \frac{n!}{(n - r)!} [/tex]
[tex]840 = \frac{n!}{(n - 4)!} [/tex]
[tex]840 = \frac{(n)(n - 1)(n - 2)(n - 3)(n - 4)!}{(n - 4)!} [/tex]
cancel (n - 4)!
[tex]840 = (n)(n - 1)(n - 2)(n - 3)[/tex]
[tex]840 = (n)(n - 1)( {n}^{2} - 5n + 6)[/tex]
[tex]840 = (n)( {n}^{3} - 5 {n}^{2} + 6n - {n}^{2} + 5n - 6)[/tex]
[tex]840 = (n)( {n}^{3} - 6 {n}^{2} + 11n - 6)[/tex]
[tex]840 = {n}^{4} - 6 {n}^{3} + 11 {n}^{2} - 6n[/tex]
[tex] {n}^{4} - 6 {n}^{3} + 11 {n}^{2} - 6n - 840 = 0[/tex]
the roots are:
[tex]n = - 4,7, \frac{3 + i \sqrt{111} }{2} , \frac{3 - i \sqrt{111} }{2} [/tex]
take the positive whole number.
n = 7
