Question

Coefficiants of (2x+y)^4​

Answer 1

By the binomial theorem,

`(2x+y)^4=\displaystyle\sum_(k=0)^4\binom 4k(2x)^(4-k)y^k=\sum_(k=0)^4\binom 4k2^(4-k)x^(4-k)y^k`

where

`\dbinom nk=(n!)/(k!(n-k)!)`

Then the coefficients of the
`x^(4-k)y^k` terms in the expansion are, in order from
`k=0` to
`k=4`,

`\dbinom 402^(4-0)=1\cdot2^4=16`

`\dbinom412^(4-1)=4\cdot2^3=32`

`\dbinom422^(4-2)=6\cdot2^2=24`

`\dbinom432^(4-3)=4\cdot2^1=8`

`\dbinom442^(4-4)=1\cdot2^0=1`