Number of Arrangements of n Objects of m Types/Examples/10 people in 3 groups sizes 5, 3, 2

Example of Use of Number of Arrangements of $n$ Objects of $m$ Types

Let $N$ be the number of ways $10$ people can be partitioned into $3$ sets: one with $5$, one with $3$ and one with $2$ people.

Then:

$N = 2520$

Proof

Here we have an instance of Number of Arrangements of $n$ Objects into $3$ Types, such that:

$n = 10$

$p = 5$

$q = 3$

$r = 2$

Hence we have:

\(\ds N\)

\(=\)

\(\ds \dfrac {10!} {5! \, 3! \, 2!}\)

Number of Arrangements of $n$ Objects into $3$ Types

\(\ds \)

\(=\)

\(\ds \dfrac {3628800} {120 \times 6 \times 2}\)

Definition of Factorial

\(\ds \)

\(=\)

\(\ds 2520\)

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $7$