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Find a positive value of m for which the coefficient of x^2 in the expansion (1 + x)^m is 6.

Q12.    Find a positive value of m for which the coefficient of x^2 in the expansion (1 + x)^ m is 6.

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As we know that the general  (r+1)^{th} term  T_{r+1} in the binomial expansion of  (a+b)^n  is given by 

T_{r+1}=^nC_ra^{n-r}b^r

So, the general  (r+1)^{th} term  T_{r+1} in the binomial expansion of  (1 + x)^ m  is

T_{r+1}=^mC_r1^{m-r}x^r=^mC_rx^r

x^2 will come when r=2. So,

The coeficient of x^2  in the binomial expansion of  (1 + x)^ m  = 6 

\Rightarrow ^mC_2=6

\Rightarrow \frac{m!}{2!(m-2)!}=6

\Rightarrow \frac{m(m-1)}{2}=6

\Rightarrow m(m-1)=12

\Rightarrow m^2-m-12=0

\Rightarrow (m+3)(m-4)=0

\Rightarrow m=4\:or\:-3

Hence the positive value of m for which the coefficient of x^2 in the expansion (1 + x)^ m is 6, is 4.

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