Hello, matty888!
Consider the experiment in which five 6-sided dice are thrown simultaneously.
How many distinguishable patterns are there, if we are interested purely
in the tally of the occurrences of each number?
I will assume that this means: the order of the numbers is not considered,
. . similar to evaluating a Poker hand.
Then there are seven possible "hands" . . .
Five-of-a-Kind
There are 6 possible values for the Five-of-a-Kind, and one way to get each of them.
. . There are: .6 Five-of-a-Kind hands.
Four-of-a-Kind
There are 6 possible values for the Four-of-a-Kind.
There are 5 possible values fo the fifth die.
. . There are: . Four-of-a-Kind hands.
Full House (Three-of-a-kind and a Pair)
There are 6 possible values for the Three-of-a-kind.
There are 5 possible values for the Pair.
. . There are: . Full Houses.
Three-of-a-Kind
There are 6 possible values for the Three-of-a-Kind.
There are: choices for the other two dice.
. . There are: . Three-of-a-Kind hands.
Two Pairs
There are: choices for the values of the Pairs.
There are: 4 choices for the value of the fifth die.
. . There are: . Two-Pair hands.
One Pair
There are 6 possible values for the one Pair.
There are: choices for the other three dice.
. . There are: . One-Pair hands.
No Pairs
There are No-Pair hands.
Therefore, there are: . possible patterns ("hands").