# Anagrams and mathematics!

Question:

How many anagrams can be formed from the word CHARACTERIZATION? (An anagram is a word having the same letters, each occurring the same number of times; this second word does not need to have a meaning).

Hint:

If we form all permutations of the letters, how often does the same word occur?

Solution:

There are $16!$ permutations of the letters of CHARACTERIZATION. However, not all of these give new words. In fact, in any permutation, if we exchange the three A’s, the two C’s, the two R’s, the two I’s or the two T’s we get the same word. Thus, for any permutation, there are $3!.2.2.2.2=96$ permutations which give the same word, so the result is

$\frac{16!}{96}$

In general, if there are $k_{A}$ A’s, $k_{B}$ B’s etc., then the result is

$\frac{(k_{A}+k_{B}+\ldots)!}{k_{A}! k_{B}! \ldots}$.

More later,

Nalin Pithwa