To determine the number of ways you can get exactly 3 sixes on the first roll of 5 dice in Yahtzee, you can use combinatorics. The formula to calculate the number of ways to choose a specific number of items from a set is called the combination formula.
In this case, you want to calculate the number of ways to select 3 out of 5 dice to be sixes. The combination formula is expressed as:
C(n, k) = n! / (k! * (n - k)!)
where n is the total number of items (in this case, the total number of dice), and k is the number of dice you want to select (in this case, the number of sixes).
So, for your specific case, the number of ways to get exactly 3 sixes on the first roll is:
C(5, 3) = 5! / (3! * (5 - 3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2 * 1)
= (5 * 4) / (2 * 1)
= 10
Therefore, there are 10 different ways to get exactly 3 sixes on the first roll of 5 dice in Yahtzee.
You do not need to use factorials to calculate this specific combination. The calculation just involves straightforward multiplication and division. However, factorials are used in the general combination formula to calculate the numerator and denominator of the formula.