Find the value of n (n is an integer).
(n+7)! / n!(n+6)(n+4) = 9!/3
Assuming the question is actually:
(n+7)! / (n!(n+6)(n+4))=9!/3
and n>0.
First simplify the left-hand side by cancelling n!:
(n+7)! / (n!(n+6)(n+4))
=(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)/((n+6)*(n+4)
=(n+1)*(n+2)*(n+3)*(n+5)*(n+7)
So the problem reduces to solving the quintic
(n+1)*(n+2)*(n+3)*(n+5)*(n+7)=9!/3
Analytic solution of the quintic is likely to be overly complex.
We can take advantage of the fact that the left hand side is strictly monotonic, and the right hand side is a constant.
Solving by binary search (of integers) gives the solution n=7 in a couple of tries.
To find the value of n, we can simplify the equation step by step.
First, we'll expand the factorial expressions:
(n + 7)! = (n + 7)(n + 6)(n + 5)(n + 4)(n + 3)(n + 2)(n + 1)n!
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Next, we'll substitute these expressions back into the original equation:
(n + 7)(n + 6)(n + 5)(n + 4)(n + 3)(n + 2)(n + 1)n! / n!(n + 6)(n + 4) = 9! / 3
Now, let's simplify further by canceling out common factors. We notice that n! appears on both the numerator and the denominator, so we can cancel them out:
(n + 7)(n + 6)(n + 5)(n + 4)(n + 3)(n + 2)(n + 1) / (n + 6)(n + 4) = 9! / 3
After canceling out common factors, we get:
(n + 7)(n + 5)(n + 3)(n + 2)(n + 1) / (n + 4) = 9 / 3
Since 9 / 3 = 3, we can substitute it into the equation:
(n + 7)(n + 5)(n + 3)(n + 2)(n + 1) / (n + 4) = 3
To find the value of n, we can multiply both sides of the equation by (n + 4) to eliminate the denominator:
(n + 7)(n + 5)(n + 3)(n + 2)(n + 1) = 3(n + 4)
Expanding both sides of the equation gives:
(n^5 + 18n^4 + 125n^3 + 420n^2 + 725n + 504) = 3n + 12
Rearranging the terms and simplifying further gives:
n^5 + 18n^4 + 125n^3 + 420n^2 + 725n + 504 = 3n + 12
Bringing all the terms to one side of the equation and simplifying gives:
n^5 + 18n^4 + 125n^3 + 420n^2 + 722n + 492 = 0
At this point, we can use numerical methods such as the Newton-Raphson method or trial and error to approximate the value of n. Unfortunately, there is no straightforward algebraic method to solve this equation for n.
Therefore, the value of n cannot be determined exactly without using numerical methods.