Question

How many six-letter permutations can be formed from the letters of the word madrid?

Answer 1

If we were to write them in a line, we have six slots and six 'distinct' letters.
Thus, the total arrangement would be 6! = 720 ways.

However, letters are just letters. Since we have two 'd's', we would have overcounted by a factor of 2!, since the two d's can change in 2! ways without us noticing.

In order to understand this further, we can develop a general pattern with repeated objects.

Let's say the word was maaddrid. We have two a's and three d's that are the same each time. Thus, the a's can interchange in 2! ways, and the d's can interchange in 3! ways.

We can further develop a general rule:
When we have p repeated elements, such as in a word, we would have overcounted by a factor of p! ways, because in the end, they will still have the same arrangements. If we have p and q repeated elements, we would have overcounted by a factor of p! AND q! ways.

Thus, when taking n objects and arranging them in a line, if we have p and q repeated elements, we give this a special formula:

$\boxed{P = \frac{n!}{p!q!}}$

Going back to the original question, we can then say that the total number of DIFFERENT arrangements becomes:
$P = \frac{6!}{2!} = \frac{720}{2} = 360$