A town has n streetlights running along the north side of ma

A town has n streetlights running along the north side of main street. The poles on which they are mounted need to be painted so that they do not rust. In how many ways may they be painted with red, white, blue, and green if an even number of them are to be painted green?

Solution

even number of them are painted green

so n/2 are painted green , this can be achieved in n/2 ways

so , the rest are painted red, white and blue this can be achieved in n/2P3 ways

total number of ways = n/2 * n/2P3

= n/2 * n/2!/(n/2-3)!

= [(n+1)/2]! / [(n-6)/2]!

= (n+1)!/(n-6)!

= (n-5) * (n-4) * (n-3) ......* n * (n+1) ways