Question

Question #557cf

Answer 1

n = 6 (which can be verified by substituting n=6 into each side of the equation).

`P(2n+4,3) = (2/3)(P(n+4,4)`
by definition of permutation and multiplying both sides by 3:
`(3)(2n+4)(2n+3)(2n+2) = (2)(n+4)(n+3)(n+2)(n+1)`

factoring out 2's from two of the right side terms:
`(3)(2)(n+2)(2n+3)(2)(n+1) = (2)(n+4)(n+3)(n+2)(n+1)`

dividing out the (n+2) and (n+1) terms on both sides:
`(3)(2)(2n+3)(2) = (2)(n+4)(n+3)`

simplifying:
`24n + 36 = 2n^2 + 14n +24`

into standard form:
`2n^2 -10n -12 = 0`

using the standard formula for quadratic solution:
`(-b +- sqrt(b^2 - 4ac))/(2a)`

= `(10 +- sqrt(100 + 96))/4`

= `(10 +- 14)/4`

= `6` or `-1` but `P(n+4,4)` is not defined for `n = -1`

So, `n = 6`