You have 20 pennies, 30 nickels, and 40 dimes. assume that the pennies are identical, the nickels are identical, and the dimes are identical. in how many ways can you put all the coins in a row?

We have 90 coins, so we have 90! ways in arranging them in 90 slots.
Since we have identical coins, we have overcounted by a factor of 20!, 30!, and 40!

We can visualise this with a portion of the coins: let's take 3 pennies, 4 nickels, and 5 dimes to demonstrate this.

We'll call them p (pennies), n  (nickels), and d (dimes).

P₁  P₂  P₃  N₁  N₂  N₃  N₄  D₁  D₂  D₃  D₄  D₅

Without the superscripts, we get the arrangement: PPPNNNNDDDD
This is one arrangement.

Now, let's try to change the position of the coins.

P₂  P₁  P₃  N₂  N₁  N₄  N₃  D₂  D₁  D₃  D₅  D₄
This is another arrangement, since the order matters. So, without the superscripts, we get:

PPPNNNNDDDDD

But we just made this arrangement in the example before.
In fact, we have this many duplicates, 20!30!40! times because the coins can change the order without us noticing.

Thus, we need to exclude these repetitions from our final number: