`(x + 1)^6` Use the Binomial Theorem to expand and simplify the expression.

We have to expand the expression using the Binomial Theorem. So use the binomial formula,

`(a+b)^n=sum_(k=0)^n((n),(k))a^(n-k)b^k`

`:.(x+1)^6=((6),(0))x^(6-0)*1^0+((6),(1))x^(6-1)*1^(1)+((6),(2))x^(6-2)*1^2+((6),(3))x^(6-3)*1^3+((6),(4))x^(6-4)*1^4+((6),(5))x^(6-5)*1^5+((6),(6))x^(6-6)*1^6`

`=x^6+(6!)/(1!(6-1)!)x^5+(6!)/(2!(6-2)!)x^4+(6!)/(3!(6-3)!)x^3+(6!)/(4!(6-4)!)x^2+(6!)/(5!(6-5)!)x^1+1`

`=x^6+(6*5!)/(5!)x^5+(6*5*4!)/(2*1*4!)x^4+(6*5*4*3!)/(3*2*1*3!)x^3+(6*5*4!)/(4!*2*1)x^2+(6*5!)/(5!)x+1`

`=x^6+6x^5+15x^4+20x^3+15x^2+6x+1`