Answer:
To find the value of ( n ) when the coefficient of ( x^3 ) is four times the coefficient of ( x^2 ) in the expansion of ( (1 + x)^n ), we set up an equation based on the binomial theorem. The relation between the coefficients is determined by the formula for the ( k^{th} ) term in the expansion of ( (a + b)^n ), which is ( \binom{n}{k}a^{n-k}b^k ).
In this case, we set up the equation where the coefficient of ( x^3 ) is 4 times the coefficient of ( x^2 ), so we have:
( \binom{n}{3} \cdot 1^{n-3} \cdot x^3 = 4 \cdot \binom{n}{2} \cdot 1^{n-2} \cdot x^2 ).
Solving this equation will give us the value of ( n ) that satisfies the condition specified.
Step-by-step explanation:
