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How many permutations of three items can be selected from a group of six? Use the letters A, B, C, D, E, and F to identify the items, and list each of the permutations of items B, D, and F.

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Answer:

In total,
120 permutations of three items can be selected from a group of six distinct elements.

In particular, there are
6 ways to order three distinct items.


\begin{aligned}\rm B-D-F \\ \rm B-F-D \\ \rm D-B-F \\ \rm D-F-B \\ \rm F-B-D \\ \rm F-D-B\end{aligned}.

Explanation:

The formula
\displaystyle P(n,\, r) = (n!)/((n - r)!) = n \, (n - 1) \cdots (n - r + 1) gives the number of ways to select and order
r items from a group of
n distinct elements.

To select and order three items from a group six distinct elements, let
n = 6 and
r = 3. Apply the formula:


\begin{aligned} P(6,\, 3) &= (6!)/((6 - 3)!) = (6!)/(3!) \\ &= (6 * 5 * 4 * 3* 2 * 1)/(3 * 2 * 1) \\ &= 6 * 5 * 4 = 120 \end{aligned}.

In other words, there are
120 unique ways to select and order three items (select a permutation of three items) from a group of six distinct elements.

Consider: what's the number of ways to order three distinct items? That's the same as asking: how many ways are there to select and order three items from a group of three distinct elements? Let
n =3 and
r = 3. Apply the formula for permutation:


\begin{aligned} P(3,\, 3) &= (3!)/((3 - 3)!) = (3!)/(0!) && \left(\text{$0! = 1$ by convention.}\right) \\ &= 3! = 3 * 2* 1 \\ &= 6\end{aligned}.

To find the permutations, start by selecting one element as the first of the list. A tree diagram might be helpful. Refer to the attachment for an example.

How many permutations of three items can be selected from a group of six? Use the-example-1
User Sebastian Hofmann
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