Bags of marbles combination/permutation problem

[Aztral]

Okie. Here goes...

Let's say you have B bags of marbles B1, B2,...Bb (all marbles are unique);
The number of marbles in each bag varies. B1 has n1 marbles, B2 has n2 marbles, etc.

You want to to create a sequences of B marbles
{marble1, marble2,...marbleb}, where marble1 came from B1, marble2 from B2, etc.

The total number of permutations is n1*n2*...nb.

But...suppose you now only allow m marbles in each permutation to be duplicated.

//////////////////Questionow many permutations are there now?//////////////////////



Ex: You have 3 bags of marbles.
B1 {red,green,blue}, n1 = 3;
B2 {white,black}, n2 = 2;
B3 {silver,gold}, n3 = 2;

I will only allow 1 marble to be duplicated, so

{red,white,silver}
{red,black,gold}
{green,white,gold}
{green,black,silver}

Can't use blue now because the options would be...
{blue,white,gold}// { x, white, gold} was used in {green,white,gold} so would have 2 duplicated marbles
{blue,white,silver}// { x, white, silver} was used in {red,white,silver} so would have 2 duplicated marbles
{blue,black,gold}// { x, black,gold} was used in {red,black,gold} so would have 2 duplicated marbles
{blue,black,silver}// { x, black, silver} was used in {green,black,silver} so would have 2 duplicated marbles



Any help would be appreciated.

Thx! :)

 

[martial]

Try to figure it out for your small example and then generalize it for your homework problem.