The coefficient of t50 in (1 + t2)25(1 + t25) (1 + t40)(1 + t45)(1 + t47) is

Question

The coefficient of t50 in (1 + t2)25(1 + t25) (1 + t40)(1 + t45)(1 + t47) is (A) 1+ 25C5 (B) 1+ 25C5+ 25C7 (C) 1+ 25C7 (D) None of these

Tardigrade Admin · Accepted Answer

(1+ 25 C1 t2+ ldots .+ 25 C25 t50) ×(1+t25+t40+t45+t47) As all the terms in the first have even exponent we can ignore t25, t45 and t47 too thus coefficient of t50 is = 25 C25+ 25 C5=1+ 25 C5

Q. The coefficient of $t^{50}$ in $(1 + t^{2})^{25} (1 + t^{25})$ $(1 + t^{40}) (1 + t^{45}) (1 + t^{47})$ is

$1 +_^{25} C_{5}$

$1 +_^{25} C_{5} +_^{25} C_{7}$

$1 +_^{25} C_{7}$

None of these

$(1 +^{25} C_{1} t^{2} + \dots . +^{25} C_{25} t^{50}) \times (1 + t^{25} + t^{40} + t^{45} + t^{47})$
As all the terms in the first have even exponent we can ignore $t^{25}, t^{45}$ and $t^{47}$ too thus coefficient of $t^{50}$ is
$=^{25} C_{25} +^{25} C_{5} = 1 +^{25} C_{5}$

Q. The coefficient of t50 in (1+t2)25(1+t25) (1+t40)(1+t45)(1+t47) is

1+_25C5​

1+_25C5​+_25C7​

1+_25C7​