Question

Write the sum of the coefficients in the expansion of $(1-3x+x^2{)}^{111}.$

Answer 1

$(1-3x+x^2{)}^{111}$

${=}^{111}{C}_0\left(1{)}^{111}{+}^{111}{C}_1\right(1{)}^{110}(-3x+x^2){+}^{111}{C}_2\left(1{)}^{109}\right(-3x+x^2{)}^2+....{+}^{111}{C}_{111}(-3x+x^2{)}^{111}$

Let x =1 on both sides.

$(1-3x+1{)}^{111}$

${=}^{111}{C}_0\left(1{)}^{111}{+}^{111}{C}_1\right(1{)}^{110}(-3+1){+}^{111}{C}_2\left(1{)}^{109}\right(-3+1{)}^2+....{+}^{111}{C}_{111}(-3+1{)}^{111}$

$(-1{)}^{111}{=}^{111}{C}_0(1{)}^{111}{+}^{111}{C}_1\left(1{)}^{110}\right(-2\left){+}^{111}{C}_2\right(1{)}^{109}(-2{)}^2+....{+}^{111}{C}_{111}(-3+1{)}^{111}$

The sum of the coefficients is $(-1{)}^{111}=-1.$