permutations and combinations

we consider (a+b)^3.

when taking care of the order of a and b,

 

(a+b)^3

= (a+b)(a+b)(a+b)

= (aa + ab + ba + bb)(a+b)

= aaa + aab + aba + abb + baa + bab + bba+ bbb

it exactly corresponds to the permutation with repetition

when taking three out of the two distinct elements(i.e. a and b) allowing for duplicates.

 

when similar terms are put together, we get

= aaa + (aab + aba + baa) + (abb + bab + bba)+ bbb

= a^3 + 3 a^2 b + 3 a b^2 + b^3

 

the coefficient of each term represents combinations.

for example, in case of 3 a^2 b,

3 means the combination choosing 2 out of the 3 distinct cards(or 1 out of the 3 distinct cards)

 

when ignoring the coefficients, we get

=>aaa , aab, abb, bbb

 

it just represents the number of the combination with repetition

when taking three out of the two distinct elements(i.e. a and b) allowing for duplicates.