The co-efficient of x3 in the expansion of (1-x+x2)5 is

Question

The co-efficient of x3 in the expansion of (1-x+x2)5 is (A) 10 (B) -20 (C) -50 (D) -30

Tardigrade Admin · Accepted Answer

No. of terms in (1 - x+ x2)5 = 1 + x(x- 1)5 = 5C0 + 5C1x(x-1)+ 5C2x2(x-1)2 + 5C3x3 (x-1)3+ 5C4x4(x-1)4+ 5C5x5(x-1)5 ∴ co - eff. x3 = - 2⋅5C2- 5C3 = -30

Q. The co-efficient of $x^{3}$ in the expansion of $(1 - x + x^{2})^{5}$ is

10

-20

-50

-30

No. of terms in
$(1 - x + x^{2})^{5} = 1 + x (x - 1)^{5}$
$=^{5} C_{0} +^{5} C_{1} x (x - 1) +^{5} C_{2} x^{2} (x - 1)^{2}$
$+^{5} C_{3} x^{3} (x - 1)^{3} +^{5} C_{4} x^{4} (x - 1)^{4} +^{5} C_{5} x^{5} (x - 1)^{5}$
$∴$ co - eff. $x^{3} = - 2 \cdot^{5} C_{2} -^{5} C_{3} = - 30$

Q. The co-efficient of x3 in the expansion of (1−x+x2)5 is

10

-20

-50

-30

No. of terms in (1−x+x2)5=1+x(x−1)5 =5C0​+5C1​x(x−1)+5C2​x2(x−1)2 +5C3​x3(x−1)3+5C4​x4(x−1)4+5C5​x5(x−1)5 ∴ co - eff. x3=−2⋅5C2​−5C3​=−30

Q. The co-efficient of $x^{3}$ in the expansion of $(1 - x + x^{2})^{5}$ is

No. of terms in
$(1 - x + x^{2})^{5} = 1 + x (x - 1)^{5}$
$=^{5} C_{0} +^{5} C_{1} x (x - 1) +^{5} C_{2} x^{2} (x - 1)^{2}$
$+^{5} C_{3} x^{3} (x - 1)^{3} +^{5} C_{4} x^{4} (x - 1)^{4} +^{5} C_{5} x^{5} (x - 1)^{5}$
$∴$ co - eff. $x^{3} = - 2 \cdot^{5} C_{2} -^{5} C_{3} = - 30$