Question

How do you use the binomial theorem to expand and simplify the expression `(x+1)^6`?

Answer 1

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

Explanation:

Binomial theorem:
`(a + b)^n =(nC_0) a^nb^0+ (nC_1) a^(n-1)b^1 +(nC_2) a^(n-2)b^2 +...... (nC_n)a^0b^n`
Here `(a=x ; b=1 , n= 6) :. 1^n=1`

We know `nC_r= (n!)/(r!(n-r)!) :. 6C_0=1 , 6C_1=6 , 6C_2=15 ,6C_3=20, 6C_4=15, 6C_5=6, 6C_6=1`

`:. (x+1)^6= x^6 + 6x^5+15x^4+20x^3+15x^2+6x+1` [Ans]