Crazy Pair of Dice...

Discrete Mathematics Level 5

Lets roll a pair of standard 6-sided dice first.

There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12.

Now, let there be a pair of another 6-sided dice A and B. (Both non-standard)

A and B satisfy these properties:

  • Each face has at least one dot.
  • The number of ways of obtaining each sum is the same as for the standard dice.

Let the number of dots on faces of A and B be represented by \(a_i\) and \(b_i\) respectively. \((i\in \{1,2,3,4,5,6\})\)

Calculate \(\displaystyle\sum_{i=1}^{6} (a_i^2+b_i^2)+\left (\displaystyle\sum_{i=1}^6 a_i\right )\left (\displaystyle\sum_{i=1}^6 b_i\right )\).


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