Question

Binomial Expansion/Pascal's triangle. Please help with all of number 5.

Answer 1

`\begin{matrix}1\\1&1\\1&2&1\\1&3&3&1\\1&4&6&4&1\end{bmatrix}`

The rows add up to
`1,2,4,8,16`, respectively. (Notice they're all powers of 2)

The sum of the numbers in row
`n` is
`2^(n-1)`.

The last problem can be solved with the binomial theorem, but I'll assume you don't take that for granted. You can prove this claim by induction. When
`n=1`,


`(1+x)^1=1+x=\dbinom10+\dbinom11x`

so the base case holds. Assume the claim holds for
`n=k`, so that


`(1+x)^k=\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^(k-1)+\dbinom kkx^k`

Use this to show that it holds for
`n=k+1`.


`(1+x)^(k+1)=(1+x)(1+x)^k`

`(1+x)^(k+1)=(1+x)\left(\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^(k-1)+\dbinom kkx^k\right)`

`(1+x)^(k+1)=1+\left(\dbinom k0+\dbinom k1\right)x+\left(\dbinom k1+\dbinom k2\right)x^2+\cdots+\left(\dbinom k{k-2}+\dbinom k{k-1}\right)x^(k-1)+\left(\dbinom k{k-1}+\dbinom kk\right)x^k+x^(k+1)`

Notice that


`\dbinom k\ell+\dbinom k{\ell+1}=(k!)/(\ell!(k-\ell)!)+(k!)/((\ell+1)!(k-\ell-1)!)`

`\dbinom k\ell+\dbinom k{\ell+1}=(k!(\ell+1))/((\ell+1)!(k-\ell)!)+(k!(k-\ell))/((\ell+1)!(k-\ell)!)`

`\dbinom k\ell+\dbinom k{\ell+1}=(k!(\ell+1)+k!(k-\ell))/((\ell+1)!(k-\ell)!)`

`\dbinom k\ell+\dbinom k{\ell+1}=(k!(k+1))/((\ell+1)!(k-\ell)!)`

`\dbinom k\ell+\dbinom k{\ell+1}=((k+1)!)/((\ell+1)!((k+1)-(\ell+1))!)`

`\dbinom k\ell+\dbinom k{\ell+1}=\dbinom{k+1}{\ell+1}`

So you can write the expansion for
`n=k+1` as


`(1+x)^(k+1)=1+\dbinom{k+1}1x+\dbinom{k+1}2x^2+\cdots+\dbinom{k+1}{k-1}x^(k-1)+\dbinom{k+1}kx^k+x^(k+1)`

and since
`\dbinom{k+1}0=\dbinom{k+1}{k+1}=1`, you have


`(1+x)^(k+1)=\dbinom{k+1}0+\dbinom{k+1}1x+\cdots+\dbinom{k+1}kx^k+\dbinom{k+1}{k+1}x^(k+1)`

and so the claim holds for
`n=k+1`, thus proving the claim overall that


`(1+x)^n=\dbinom n0+\dbinom n1x+\cdots+\dbinom n{n-1}x^(n-1)+\dbinom nnx^n`

Setting
`x=1` gives


`(1+1)^n=\dbinom n0+\dbinom n1+\cdots+\dbinom n{n-1}+\dbinom nn=2^n`

which agrees with the result obtained for part (c).