Want to enhance this question? Upday the question so it's on-topic for jiyuushikan.orgematics Stack Exchange.

Closed 6 years ago.

You are watching: How many ways are there to rearrange the letters in function?

Using all the letters of the word ARRANGEMENT how many type of various words utilizing all letters at a time have the right to be made such that both A, both E, both R both N happen together .



$egingroup$ In basic if you have $n$ objects via $r_1$ objects of one sort, $r_2$ objects of another,...,and also $r_k$ objects of the $k$th type, they can be arranged in $$fracn!(r_1!)(r_2!)dots(r_k!)$$ ways. $endgroup$
"ARRANGEMENT" is an eleven-letter word.

If tbelow were no repeating letters, the answer would certainly ssuggest be $11!=39916800$.

See more: Why Does My Dog Stretch All The Time, Why Does My Dog Stretch So Much

However before, given that there are repeating letters, we have to divide to remove the duplicates appropriately.Tright here are 2 As, 2 Rs, 2 Ns, 2 Es

Therefore, tright here are $frac11!2!cdot2!cdot2!cdot2!=2494800$ means of arranging it.


Words ARRANGEMENT has $11$ letters, not all of them distinctive. Imagine that they are created on little bit Scrabble squares. And expect we have actually $11$ consecutive slots right into which to put these squares.

Tbelow are $dbinom112$ means to choose the slots wbelow the 2 A"s will go. For each of these means, tright here are $dbinom92$ means to decide where the two R"s will go. For eextremely decision about the A"s and also R"s, tright here are $dbinom72$ means to decide wbelow the N"s will go. Similarly, tbelow are currently $dbinom52$ ways to decide wbelow the E"s will go. That leaves $3$ gaps, and $3$ singleton letters, which deserve to be arranged in $3!$ ways, for a total of $$inom112inom92inom72inom523!.$$


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

Not the answer you're looking for? Browse other questions tagged permutations or ask your own question.

In exactly how many kind of means can the letters of the word 'arrange' be arranged if the 2 r's and the two a's do not happen together?
In how many means deserve to the letters of word $PERMUTATIONS$ be arranged if tright here are constantly 4 letters in between P and S?

website architecture / logo design © 2021 Stack Exadjust Inc; user contributions licensed under cc by-sa. rev2021.9.2.40142

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange have the right to keep cookies on your tool and discshed information in accordance through our Cookie Policy.