If nP3 - 6(nC4) = 0, find the value of n

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.