If the coefficient of x78 in the expansion of (1+x+2 x2+4 x4)20 is λ ⋅ 240 then λ is equal to

Question

If the coefficient of x78 in the expansion of (1+x+2 x2+4 x4)20 is λ ⋅ 240 then λ is equal to (A) 11 (B) 10 (C) 8 (D) 4

Tardigrade Admin · Accepted Answer

(1+ x +2 x 2+4 x 4)20 T r +1= 20 C r ⋅(1+ x +2 x 2)20- r ⋅(4 x 4) r text For r =19 T 19+1= 20 C 19 ⋅(1+ x +2 x 2)1 ⋅(4 x 4)19 ∴ Coefficient of x78=2 ⋅ 20 C 10 ⋅ 419=2 ⋅ 20 ⋅ 238=10 ⋅ 240

Q. If the coefficient of x78 in the expansion of (1+x+2x2+4x4)20 is λ⋅240 then λ is equal to

11

10

8

4

(1+x+2x2+4x4)20 Tr+1​=20Cr​⋅(1+x+2x2)20−r⋅(4x4)r For r=19 T19+1​=20C19​⋅(1+x+2x2)1⋅(4x4)19 ∴ Coefficient of x78=2⋅20C10​⋅419=2⋅20⋅238=10⋅240

Q. If the coefficient of $x^{78}$ in the expansion of $(1 + x + 2 x^{2} + 4 x^{4})^{20}$ is $λ \cdot 2^{40}$ then $λ$ is equal to