6(n+1P2)=3nP2

To solve the equation 6(n+1)P2 = 3nP2, we first need to understand what is meant by "P2." It is likely that "P2" refers to the concept of "permutation of 2 objects."

A permutation of 2 objects represents the number of ways you can arrange those 2 objects. In this case, it seems we are dealing with the permutation of 2 objects, denoted as P2. The formula for a permutation is given by:

nP2 = n! / (n - r)!

Where n is the total number of objects and r is the number of objects you want to arrange (in this case, 2).

Now, let's substitute the P2 in our equation with the permutation formula:

6(n+1)! / ((n+1) - 2)! = 3n! / (n - 2)!

Next, simplify the factorial expressions:

6(n+1)! / (n-1)! = 3n! / (n-2)!

Notice that both sides of the equation have n! terms. We can multiply both sides of the equation by (n-2)! to eliminate it:

6(n+1)! = 3(n-1)!

Now, we can simplify further:

6(n+1)(n)(n-1)! = 3(n-1)!

Divide both sides of the equation by (n-1)! to isolate the factor (n-1):

6(n+1)(n) = 3

Simplify the equation:

6n^2 + 6n - 18 = 0

Now, we have a quadratic equation. We can either factor it or use the quadratic formula to find the values of n that satisfy the equation.