Question

Find the sum of all coefficients in the following binomial expansion.

a. (2u + v)^10
b. (2u − v)^10
c. (2u − 3v)^11
d. (u − 3v)^11
e. (1 + i)^10
f. (1 − i)^10
g. (1 + i)^200
h. (1 + i)^201

Answer 1

We can find the sum of all the coefficients by substituting all the variables in the expansion with one.

a. u=1,v=1

sum=
`(2*1+1)^(10)`=
`3^(10)`

b.u=1,v=1

sum=
`(2*1-1)^(10)`=1

c.u=1,v=1

sum=
`(2*1-3*1)^{11`=-1

d.u=1,v=1

sum=
`(1-3*1)^(11)`=-
`2^{11`

e.i=1

sum=
`(1+1)^{10`=
`2^{10`

f.i=1

sum=
`(1-1)^{10`=0

g.i=1

sum=
`(1+1)^{200`=
`2^{200`

h.i=1

sum=
`(1+1)^(201)`=
`2^{201`