**Want to enhance this question?**upgrade the inquiry so it's on-topic for jajalger2018.org Stack Exchange.

Closed 6 year ago.

You are watching: How many ways to arrange letters in a word

Using every the letters of the word arrangement how countless different words using all letter at a time deserve to be make such the both A, both E, both R both N take place together .

$egingroup$ In general if you have actually $n$ objects v $r_1$ objects that one kind, $r_2$ objects the another,...,and $r_k$ objects the the $k$th kind, they deserve to be arranged in $$fracn!(r_1!)(r_2!)dots(r_k!)$$ ways. $endgroup$

"ARRANGEMENT" is an eleven-letter word.

If there to be no repeating letters, the answer would just be $11!=39916800$.

See more: What Does The Suffix Ment Mean, Ment Meaning

However, because there are repeating letters, we have to divide to remove the duplicates accordingly.There room 2 As, 2 Rs, 2 Ns, 2 Es

Therefore, there are $frac11!2!cdot2!cdot2!cdot2!=2494800$ ways of arranging it.

The word arrangement has $11$ letters, not all of them distinct. Imagine that they room written on little Scrabble squares. And also suppose we have $11$ consecutive slots into which to placed these squares.

There room $dbinom112$ methods to **choose** the slots where the two A"s will go. Because that **each** of these ways, there room $dbinom92$ methods to decide wherein the 2 R"s will go. For every decision about the A"s and also R"s, there room $dbinom72$ methods to decide wherein the N"s will go. Similarly, there are now $dbinom52$ means to decide where the E"s will go. That pipeline $3$ gaps, and $3$ singleton letters, which have the right to be i ordered it in $3!$ ways, because that a total of $$inom112inom92inom72inom523!.$$

Highly active question. Earn 10 call (not counting the combination bonus) in order come answer this question. The reputation requirement helps protect this question from spam and also non-answer activity.

## Not the answer you're spring for? Browse various other questions tagged permutations or asking your very own question.

In how many ways have the right to the letters of words 'arrange' be i ordered it if the two r's and the two a's carry out not take place together?

In how numerous ways deserve to the letter of word $PERMUTATIONS$ be i ordered it if there are constantly 4 letters between P and S?

site architecture / logo design © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.10.25.40556

her privacy

By click “Accept all cookies”, girlfriend agree stack Exchange have the right to store cookies on your machine and disclose information in accordance with our Cookie Policy.