Calculate permutations with repeated elements

This skill involves determining the number of distinct linear arrangements of a finite collection containing repeated, indistinguishable elements, using $\frac{n!}{n_1!n_2!\cdots n_k!}$ , where $n$ is the total number of elements and $n_i$ are the multiplicities of each repeated type. The reasoning accounts for overcounting that occurs when identical elements are temporarily treated as distinct; the scope is ordinary unrestricted arrangements, not circular permutations, positional restrictions, or advanced enumeration methods.

Detailed Explanation: 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.

Learn by doing: Calculate permutations with repeated elements

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Probability Counting - Duplicate Orders in 4 Letters, 1 Repeat - to Answer

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Calculate permutations with repeated elements

Detailed Explanation: Calculate permutations with repeated elements

Learn by doing: Calculate permutations with repeated elements

Practice with unlimited practice problems

Practice:

Probability Counting - Duplicate Orders in 4 Letters, 1 Repeat - to Answer