The given expression is an equation that involves the binomial coefficients nP3 and nC4, where n is an integer. We are asked to find the value of n that satisfies the equation:
nP3 - 6(nC4) = 0
Using the formula for the binomial coefficients, we can simplify the expression:
nP3 = n(n-1)(n-2)
nC4 = n(n-1)(n-2)(n-3)/4!
Substituting these expressions into the given equation, we get:
n(n-1)(n-2) - 6(n(n-1)(n-2)(n-3)/4!) = 0
Simplifying further:
n(n-1)(n-2)(1 - 6(n-3)/4!) = 0
Since n is an integer, we must have either n = 0, n = 1, n = 2 or 1 - 6(n-3)/4! = 0. However, n cannot be 0, 1, or 2 because nP3 and nC4 are defined only for n >= 4.
Therefore, we need to solve the equation:
1 - 6(n-3)/4! = 0
6(n-3)/4! = 1
n-3 = 4! / 6
n-3 = 4
n = 7
Hence, the value of n that satisfies the given equation is 7.