Binomial TheoremHard
Question
If in the expansion of (1+ x)m (1 - x)n, the coefficients of x and x2 are 3 and - 6respectively, then m is
Options
A.6
B.9
C.12
D.24
Solution
(1 + x)m (1 - x)n = 
= 1 + (m - n)x +
x2 + ....
term containing power of x ≥ 3.
Now, m - n = 3 ....(i)
(∵ coefficient of x = 3 given)
and
m(m - 1) + n(n - 1) - mn = - 6
⇒ m(m -1) + n(n -1) - 2mn = - 12
⇒ m2 - m + n2 - n - 2mn = - 12
⇒ (m - n)2 - (m + n) = - 12
⇒ m+ n = 9 + 12 = 21 ......(ii)
On solving Eqs. (i) and (ii) , we get
m = 12
= 1 + (m - n)x +
x2 + .... term containing power of x ≥ 3.
Now, m - n = 3 ....(i)
(∵ coefficient of x = 3 given)
and
m(m - 1) + n(n - 1) - mn = - 6⇒ m(m -1) + n(n -1) - 2mn = - 12
⇒ m2 - m + n2 - n - 2mn = - 12
⇒ (m - n)2 - (m + n) = - 12
⇒ m+ n = 9 + 12 = 21 ......(ii)
On solving Eqs. (i) and (ii) , we get
m = 12
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