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Question: If in the expansion of $(1 + x)^m(1 - x)^n$, the coefficients of x and $x^2$ are 2 and -3, respectiv...

If in the expansion of (1+x)m(1−x)n(1 + x)^m(1 - x)^n, the coefficients of x and x2x^2 are 2 and -3, respectively then 2m−n2m - n equals

A

8

B

6

C

4

D

2

Answer

8

Explanation

Solution

To find the value of 2m−n2m - n, we first need to determine the values of mm and nn using the given information about the coefficients of xx and x2x^2 in the expansion of (1+x)m(1−x)n(1 + x)^m(1 - x)^n.

The binomial expansions are: (1+x)m=1+mx+m(m−1)2!x2+O(x3)(1 + x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + O(x^3) (1−x)n=1−nx+n(n−1)2!x2+O(x3)(1 - x)^n = 1 - nx + \frac{n(n-1)}{2!}x^2 + O(x^3)

Now, let's multiply these two series to find the terms up to x2x^2: (1+x)m(1−x)n=(1+mx+m(m−1)2x2+… )(1−nx+n(n−1)2x2+… )(1 + x)^m(1 - x)^n = \left(1 + mx + \frac{m(m-1)}{2}x^2 + \dots\right) \left(1 - nx + \frac{n(n-1)}{2}x^2 + \dots\right)

To find the coefficient of xx: Coefficient of x=(1)(−n)+(m)(1)=m−nx = (1)(-n) + (m)(1) = m - n

Given that the coefficient of xx is 2, we have: m−n=2(Equation 1)m - n = 2 \quad \text{(Equation 1)}

To find the coefficient of x2x^2: Coefficient of x2=(1)(n(n−1)2)+(mx)(−nx)+(m(m−1)2)(1)x^2 = (1)\left(\frac{n(n-1)}{2}\right) + (mx)(-nx) + \left(\frac{m(m-1)}{2}\right)(1) Coefficient of x2=n(n−1)2−mn+m(m−1)2x^2 = \frac{n(n-1)}{2} - mn + \frac{m(m-1)}{2}

Given that the coefficient of x2x^2 is -3, we have: n(n−1)2−mn+m(m−1)2=−3\frac{n(n-1)}{2} - mn + \frac{m(m-1)}{2} = -3 Multiply the entire equation by 2 to clear the denominators: n(n−1)−2mn+m(m−1)=−6n(n-1) - 2mn + m(m-1) = -6 n2−n−2mn+m2−m=−6n^2 - n - 2mn + m^2 - m = -6

Rearrange the terms to group m2,−2mn,n2m^2, -2mn, n^2: (m2−2mn+n2)−(m+n)=−6(m^2 - 2mn + n^2) - (m + n) = -6 This simplifies to: (m−n)2−(m+n)=−6(m-n)^2 - (m+n) = -6

Substitute the value of (m−n)(m-n) from Equation 1 into this equation: (2)2−(m+n)=−6(2)^2 - (m+n) = -6 4−(m+n)=−64 - (m+n) = -6 m+n=4+6m+n = 4 + 6 m+n=10(Equation 2)m+n = 10 \quad \text{(Equation 2)}

Now we have a system of two linear equations:

  1. m−n=2m - n = 2
  2. m+n=10m + n = 10

Add Equation 1 and Equation 2: (m−n)+(m+n)=2+10(m - n) + (m + n) = 2 + 10 2m=122m = 12 m=6m = 6

Substitute the value of m=6m=6 into Equation 1: 6−n=26 - n = 2 n=6−2n = 6 - 2 n=4n = 4

Finally, we need to calculate 2m−n2m - n: 2m−n=2(6)−42m - n = 2(6) - 4 2m−n=12−42m - n = 12 - 4 2m−n=82m - n = 8