The game of Yahtzee is played with five fair six-sided dice. A turn consists of rolling all five dice, selecting the ones you want to keep, re-rolling the rest, and then doing this one more time, if necessary. You need to match certain conditions in order to score points.

(For convenience, you may assume that the dice are all different colors.)

Suppose you roll the 5 dice seeking to get as many sixes as possible. how many ways can you get exactly 3 sixes on the first roll?

How would I do this? Factorials?
Please help.

For convenience you may assume the dice are different colors. Suppose you roll the 5 dice seeking to get as many sixes as possible. how many ways can you get exactly 3 sixes on the first roll?

How many ways can you get at least 3 sixes on the first roll?
Suppose you only got 1 six on the first roll so you keep it and roll the remaning 4 dice. how many ways can you get at least one six on the last roll?

Please help. is it factorials?

a) SSSNN

SSNSN
SSNNS
SNSSN
SNSNS
SNNSS
NSSSN
NSSNS
NSNSS
NNSSS
Are there any others?
Is this the same as 5!/3!2!

To determine the number of ways you can get exactly 3 sixes on the first roll in Yahtzee, we can use a combination formula.

In each roll, there are two possibilities for each dice: either it is a six or it is not. Since you're seeking to get exactly 3 sixes, that means you want to select 3 dice to show six and 2 dice not to show six.

The combination formula is used to calculate the number of ways to choose a specific number of items from a larger set without considering their order. In this case, we want to select 3 dice out of the 5 total dice to show six, so the combination formula will be used.

The combination formula is given by:

C(n, r) = n! / (r! * (n - r)!)

Where:
- n represents the total number of items (5 dice)
- r represents the number of items to be chosen (3 sixes)

Applying this formula to our specific case, we have:

C(5, 3) = 5! / (3! * (5 - 3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2!)
= (5 * 4) / 2!
= 20 / 2
= 10

Therefore, there are 10 ways to get exactly 3 sixes on the first roll in Yahtzee.

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.