Question

X+1 ^ 16 what is the coefficient of x^5

Answer 1

Binomial theorem:


`(x+1)^(16)=\displaystyle\sum_(k=0)^(16)\dbinom{16}kx^(16-k)1^k=\sum_(k=0)^(16)\dbinom{16}kx^(16-k)`

The
`x^5` term corresponds with the term in the sum for which
`16-k=5\implies k=11`. For this
`k`, we get


`\dbinom{16}{11}x^(16-11)=\dbinom{16}{11}x^5`

So the coefficient you want to find is


`\dbinom{16}{11}=(16!)/(11!(16-11)!)=4368`