For non$-$negative integers k$,$ n$,$m such that $0$ k $$ n $+$ m$,$ we have

$$n $+$ m

k

$=$

k

$!$

j$=0$

$$n

j

$$ m

k $$ j

$$

$.$

Here we use the convention that $$i

l$=0$ if l $>$ i$.$

c$.$ For every n in N$,$ n$2=2$n$2$

$+$ n$.$

each identity, provide two proofs, an algebraic and a combinatorial

proof.

a$.$ For every integer $0$ j $$ k $$ n$,$nk

$$kj

$=$nj

$$n$$j

k$$j$.$

b$.$ For non$-$negative integers k$,$ n$,$m such that $0$ k $$ n $+$ m$,$ we have

$$n $+$ m

k

$=$

k

$!$

j$=0$

$$n

j

$$ m

k $$ j

$$

$.$

Here we use the convention that $$i

l$=0$ if l $>$ i$.$