Solve for n given: nP2 = 110
nP2=110
To solve the equation nP2 = 110, we need to find the value of n. In nP2, P represents the permutation function.
The permutation function nP2 is defined as n! / (n-2)!, where ! represents a factorial.
So, we have the equation n! / (n-2)! = 110.
To solve this equation step-by-step, we can start by expanding the factorial expressions:
n!(n-2)! = (n)(n-1)(n-2)!
Now, we can divide both sides of the equation by (n-2)!:
n! = 110(n-2)!
Next, we can divide both sides of the equation by (n-2)! again to cancel it out:
(n-2)! = n! / 110
Now, we need to find a numerical value for n that satisfies this equation. We can do this by trying different values of n and calculating the factorial values until we find a result that gives us a factorial of 110:
n=5: (5-2)! = 5! / 110 -> 3! = 120 / 110 -> 6 = 1.09 (not equal)
n=6: (6-2)! = 6! / 110 -> 4! = 720 / 110 -> 24 = 6.54 (not equal)
n=7: (7-2)! = 7! / 110 -> 5! = 5040 / 110 -> 120 = 45.81 (not equal)
n=8: (8-2)! = 8! / 110 -> 6! = 40320 / 110 -> 720 = 366.54 (not equal)
After trying different values, we see that none of them satisfy the equation n!(n-2)! = 110.
Therefore, there is no whole number solution for n that makes nP2 equal to 110.
To solve for n in the equation nP2 = 110, we need to understand what the notation nP2 represents. The "P" stands for the permutation, and the number 2 represents the number of elements taken at a time.
The permutation formula, which calculates the number of ways to arrange n elements taken r at a time, is given by:
nP2 = n! / (n - r)!
In this equation, n! represents the factorial of n, which is the product of all positive integers from 1 to n.
By substituting nP2 = 110 into the permutation formula, we can solve for n:
n! / (n - 2)! = 110
To simplify the equation, we can expand the factorials:
(n * (n - 1) * (n - 2)! ) / (n - 2)! = 110
Since (n - 2)! appears in both the numerator and denominator, we can cancel them out:
n * (n - 1) = 110
Expanding the left side of the equation gives us a quadratic equation:
n^2 - n = 110
Rearranging the terms:
n^2 - n - 110 = 0
Now, we can solve this quadratic equation for n. There are different methods to solve it, such as factoring, completing the square, or using the quadratic formula. Let's solve it by factoring:
(n + 10)(n - 11) = 0
Setting each factor equal to zero:
n + 10 = 0 or n - 11 = 0
Solving each equation gives us two possible values for n:
n = -10 or n = 11
Therefore, there are two possible solutions for n: n = -10 and n = 11.