How many possible outcomes are their when you roll five six sided dice?

Answer 1

Here are two ways you can solve this, depending on if order matters or order does not matter:

Order of the dice does matter:

Assuming that you roll a 1, 1, 1, 1, 4 is not the same as rolling a 1, 4, 1, 1, 1

Think of 5 blanks first (one for each six sided dice).[ dice 1 ] [ dice 2 ] [ dice 3 ] [ dice 4 ] [ dice 5 ]

Each dice has 6 sides, so you can have 6 different possibilities per dice. Fill in each blank with how manypossibilities. Then multiply them.

(6) x (6) x (6) x (6) x (6) = 65 = 7776 possibilities.

A few possibilities looks like (I'm not listing all 7776possibilities):

(1, 1, 1, 1, 1), (2, 1, 1, 1, 1), (3, 1, 1, 1, 1), (4, 1, 1, 1, 1), (5, 1, 1, 1, 1), (6, 1, 1, 1, 1), where (1, 1, 1, 1, 1) represents all five dice landing on a 6.

The answer is 7776 possibilities.

Order of the dice does not Matter

Now if 1, 1, 1, 1, 4 is the same as rolling a 1, 4, 1, 1, 1 then the problem is more complex. You can use the following formula (it's a binomial):

C(n+5, 5), where n is the amount of dice

C(5+5,5) = C(10,5)

How to use this:

C(A,B) = A! / [ (A-B)! B! ], where A! is read A factorial

What is a factorial:

n! = n x (n-1) x (n-2) ... x ... (n-n+1)

Example:

5! = 5 x (5-1) x (5-2) x (5-3) x (5-4)

5! = 5 x 4 x 3 x 2 x 1

Plug in your numbers;

C(10,5) = 10! / [ (10-5)! 5! ]

C(10,5) = 10! / ( 5! 5! )

( 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 ) / ( 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 x 5 x 5 )

= ( 6 x 7 x 8 x 9 x 10 ) / ( 2 x 3 x 4 x 5 ) = 30240 / 120 = 252 possibilities

The equation I gave you works for six sided dice problems. I have verified this answer numerically.

So the answers are:

7776 possibilities if order matters.

252 possibilities if order does not matter