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A Dice Combination Problem

If you have two 6-sided dice, numbered 1-6, how many dice combinations are there that do not include the number 5?

Note that a 1 and 6 and a 6 and 1 with the two dice are counted as separate dice combinations.

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There are six sides to each die, so to find the total number of dice combinations – multiply 6 x 6 – 36 total combinations. Now for each number on die A, there is a combination where die B is 5 – so that is 6 combinations to start with. Now, to account for the dice combinations where die A is 5, there are six possibilities on die 2. Since we have already accounted for the combination where die B is 5, there are only 5 more to add. 6 + 5 = 11 dice combinations that include a 5. Subtract that from 36 – and there are 25 dice combinations that do not include a 5.

A simple approach to this problem is to reduce the number of possibilities by reducing the number of numbers on each die by 1, making it a 5 x 5 problem – 25 possible dice combinations that do not include 5.