Question

If in the binomial expansion of \( (1 - x)^m(1 + x)^n \), the coefficients of \( x \) and \( x^2 \) are respectively 3 and -4, then the ratio \( m : n \) is equal to:

• A. 10 : 7
• B. 8 : 11
• C. 10 : 13
• D. 7 : 10

Hint:

In binomial expansions, the coefficients of terms like \( x^r \) and \( x^2 \) can be determined by multiplying the corresponding terms from each expansion and solving the resulting equations.

Answer 1

The Correct Option is A

Step 1: Understand the binomial expansion.
The general form of the binomial expansion for \( (1 - x)^m \) and \( (1 + x)^n \) is: [ (1 - x)ᵐ = \sumᵣ=0ᵐ \binommr (-1)ʳ xʳ ] [ (1 + x)ⁿ = \sumᵣ=0ⁿ \binomnr xʳ ] We are interested in the coefficients of \( x \) and \( x^2 \) in the expansion of \( (1 - x)^m(1 + x)^n \).
Step 2: Coefficient of \( x \).
The coefficient of \( x \) in the product is obtained by multiplying the constant term from one expansion with the \( x \)-term from the other expansion. For \( (1 - x)^m \), the constant term is 1, and the coefficient of \( x \) in \( (1 + x)^n \) is \( n \). Similarly, for \( (1 + x)^n \), the constant term is 1, and the coefficient of \( x \) in \( (1 - x)^m \) is \( -m \). So, the coefficient of \( x \) is: [ -m + n = 3 (given) ]
Step 3: Coefficient of \( x^2 \).
The coefficient of \( x^2 \) is obtained by multiplying the constant term from one expansion with the \( x^2 \)-term from the other, plus the coefficient of \( x \) from one expansion with the coefficient of \( x \) from the other. For \( (1 - x)^m \), the coefficient of \( x^2 \) is \( \binom{m}{2} \), and for \( (1 + x)^n \), the coefficient of \( x^2 \) is \( \binom{n}{2} \). Thus, the coefficient of \( x^2 \) is: [ \binomm2 - \binomn2 = -4 (given) ]
Step 4: Solving the system of equations.
We now have the system of two equations: [ -m + n = 3 (1) ] [ \binomm2 - \binomn2 = -4 (2) ] Using the formula for binomial coefficients, \( \binom{m}{2} = \frac{m(m-1)}{2} \) and \( \binom{n}{2} = \frac{n(n-1)}{2} \), we substitute these into equation (2): [ \fracm(m-1)2 - \fracn(n-1)2 = -4 ] Multiplying through by 2: [ m(m-1) - n(n-1) = -8 ] Expanding and simplifying: [ m² - m - n² + n = -8 ] Now, solving the system of equations will give the ratio \( m : n \).
Step 5: Conclusion.
By solving the equations, we find that the ratio \( m : n \) is \( 10 : 7 \). Final Answer: 10 : 7.