Binomial TheoremHard
Question
If (1 + 2x + 3x2)10 = a0 + a1x + a2x2 +.... + a20x20, then :
Options
A.a1 = 20
B.a2 = 210
C.a4 = 8085
D.a20 = 22. 37. 7
Solution
General term = 
(1)r1 (2x)r2 (3x2)r3
a1 = Coeff. of x
r2 + 2r3 = 1 ⇒ r2 = 1, r1 = 9, r3 = 0
∴ a1 =
(2)1 = 20
a2 = Coeff. of x2
r2 + 2r3 = 2 ⇒ r2 = 2, r1 = 8, r3 = 0
r2 = 0, r1 = 9, r3 = 1
a2 =
(2)2 + 
(3) = 210
a4 = coeff. of x4
r2 + 2r3 = 4 ⇒ r2 = 4, r1 = 6, r3 = 0
r2 = 2, r1 = 7, r3 = 1
r2 = 0, r1 = 8, r3 = 2
a4 =
(2)4 + 
(2)2 (3) + 
(3)2
= 8085
a20 = 310
(1)r1 (2x)r2 (3x2)r3a1 = Coeff. of x
r2 + 2r3 = 1 ⇒ r2 = 1, r1 = 9, r3 = 0
∴ a1 =
(2)1 = 20a2 = Coeff. of x2
r2 + 2r3 = 2 ⇒ r2 = 2, r1 = 8, r3 = 0
r2 = 0, r1 = 9, r3 = 1
a2 =
(2)2 + (3) = 210a4 = coeff. of x4
r2 + 2r3 = 4 ⇒ r2 = 4, r1 = 6, r3 = 0
r2 = 2, r1 = 7, r3 = 1
r2 = 0, r1 = 8, r3 = 2
a4 =
(2)4 + (2)2 (3) + (3)2= 8085
a20 = 310
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