Skill: Calculate permutations with repeated elements

When some elements are identical, use

$$\frac{n!}{n_1!n_2!\cdots n_k!},$$

where:

• $n$ is the total number of elements,
• $n_1,n_2,\ldots,n_k$ are the numbers of repeated copies of each type.

The denominator removes overcounting. If identical elements were temporarily labeled, such as $L_1$ and $L_2$, swapping them would not create a new arrangement. Dividing by $2!$ corrects for these duplicate arrangements.

Example: How many distinct arrangements can be made from the letters in BALLOON?

Step 1: Count all letters.

There are $7$ letters, so $n=7$.

Step 2: Identify repeated letters.

• $L$ appears $2$ times.
• $O$ appears $2$ times.
• $B$, $A$, and $N$ each appear once.

Step 3: Substitute into the formula.

The repeated letters contribute $2!$ for the Ls and $2!$ for the Os:

$$\text{Number of arrangements} = \frac{7!}{2!2!}.$$

Step 4: Calculate.

$$\frac{7!}{2!2!} = \frac{5040}{2\cdot 2} = \frac{5040}{4} = 1260.$$

Therefore, there are

$$\boxed{1260}$$

distinct linear arrangements of the letters in BALLOON.