Dice Sum Probability

You roll dice fair dice, each with faces numbered 1 to sides. What is the probability that the faces add up to exactly total?

Write dice_probability(dice, sides, total) that returns the exact probability as a string 'p/q' in lowest terms. An impossible total gives '0/1', and a certain one gives '1/1'.

Examples

dice_probability(2, 6, 7)     # returns '1/6'
dice_probability(3, 6, 10)    # returns '1/8'
dice_probability(1, 6, 7)     # returns '0/1'

Two ordinary dice make 7 in 6 of their 36 equally likely outcomes, so the answer is 6/36 = 1/6.

Constraints

• 1 <= dice <= 100
• 1 <= sides <= 100
• 0 <= total <= 10,000
• With 100 dice there are far too many outcomes to list, and the top and bottom of the fraction can have hundreds of digits.