Number of Arrangements of n Objects of m Types/Examples/Letters in added

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

Let $N$ be the number of different arrangements of the letters in the word $\texttt{added}$.

Then:

$N = 20$

Proof

There are $3$ types of letter in $\texttt{added}$, that is:

$\texttt a$: $1$ instance

$\texttt d$: $3$ instances

$\texttt e$: $1$ instance

Hence we have an instance of Number of Arrangements of $n$ Objects of $m$ Types, such that:

$n = 5$

$k_1 = 1$

$k_2 = 3$

$k_3 = 1$

Hence we have:

$N = \dfrac {5!} {1! \times 3! \times 1!} = \dfrac {120} 6 = 20$

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $2$