$($b$)$ Find all jin$\{0,1,$dots,$n\}$ such that for all $0in,$

$([n],[j])([n],[i])$

$($c$)$ Take $j_1,j_2,j_{3}in\{0,1,$dots,$n\}$ such that for all $0i_1,i_2,i_{3}n,i_1+i_2+i_3=n,$

$([n],[j_1,j_2,j_3])([n],[i_1,i_2,i_3]).$

Using part $(b)$ and

$([n],[i_1,i_2,i_3])=([n],[i_1])([n-i_1],[i_2])$

show that

$j_3{j}_2+1$

$($d$)$ Determine $j_1,j_2,j_3$ in part $(c).$

Optional $($ungraded$)$: for a given kinN, find all tuples $(j_1,j_2,$dots,$j_k)$ with

$([n],[j_1,j_2,$dots,$j_k])([n],[i_1,i_2,$dots,$i_k]).$

for all $0i_1,i_2,$dots,$i_{k}n,i_1+i_2+$dots$+i_k=n.$

Note: The answers to problem $2$ can be guessed by either computing a few by hand or writing a program

finding the answers for small values of $n,$ and then justified in a general setting. However, the formula in

problem $3$ is $($slightly$)$ more difficult to tackle with this hack.

$($b$)($c$)$and$($d$)$please$!$