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wisterville · #2 August 21st, 2008, 01:13 AM

Hello,

You divide 20 (identical) dices into 3 categories: 1, 2 or 3.

So,
${}_3H_{20}={}_{3+20-1}C_{20}={}_{22}C_{20}={}_{22}C_2=\frac{22\cdot 21}{2!}=231.$

Maybe this notation is not so common, so let me explain a bit.
${}_{22}C_{20}$ is the binomial coefficient, sometimes written $\begin{pmatrix}22\\20\end{pmatrix}$ .
It counts the combinations, allowing no repetitions.
(You choose 20 different elements out of 22 ones. The order of choice is arbitrary.)
${}_{3}H_{20}$ counts the combinations, allowing repetitions.
(You choose 20 (maybe same)elements out of 3 ones. The order of choice is arbitrary.)

You align the 20 dices and (3-1=)2 separators in one row.
The dices left of the separators are 1,
the dices between the separators are 2,
the dices right of the separators are 3.
How many alignments are there?
You have 22 places to put the dices or separators,
choose 20 places for the dices and you are done.

Bye.

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