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 Math Central Quandaries & Queries
 Question from Jacob: I need to know all the possible 3 digit combinations from the following pool of digits: 1st pool (1,2,3), 2nd pool (4,5) and the 3rd pool (7,8,9,0). Each pool has to be represented in the combinations. The numbers can be repeated as long as they are not of the same set. Example for not repeating: 123 is ok but not 321 or 231,132,213, etc. The question is somehow related to Mike's but the difference is the digits are allocated into pools. Your help can assist me solve a mystery. Thanks.

Hi Jacob,

Since order is not important I am going to write a three digit combination by first choosing an item from the first pool, then an item from the second pool and lastly an item from the third pool. There are three items in the first pool so I have 3 possibilities in choosing an item from the first pool. There are two items in the second pool so no matter which item I chose from the first pool I have 2 possibilities for choosing an item from the second pool. Hence I have $3 \times 2$ possible choices for an item from the first pool and an item from the second pool.

In a similar fashion regardless of which of the $3 \times 2$ choices you have made from pools one and two you have four choices for an item from the third pool. Hence there are $3 \times 2 \times 4$ possible combinations from the three pools.

I hope this helps,
Penny