Want to boost this question? Upday the question so it's on-topic for brickandmortarphilly.comematics Stack Exreadjust.

Closed 6 years earlier.

You are watching: How many different arrangements of the letters in the word better are there?

Using all the letters of the word ARRANGEMENT exactly how many various words making use of all letters at a time can be made such that both A, both E, both R both N occur together .



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

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

See more: Watch Billions Project Free Tv, Billions Season 4 Episode 5

However before, because there are repeating letters, we have to divide to remove the duplicates as necessary.Tright here are 2 As, 2 Rs, 2 Ns, 2 Es

Because of this, tright here are $frac11!2!cdot2!cdot2!cdot2!=2494800$ ways of arranging it.


Words ARRANGEMENT has $11$ letters, not every one of them distinct. Imagine that they are written on bit Scrabble squares. And mean we have actually $11$ consecutive slots right into which to put these squares.

There are $dbinom112$ ways to choose the slots where the 2 A"s will certainly go. For each of these methods, there are $dbinom92$ means to decide wbelow the 2 R"s will certainly go. For eincredibly decision about the A"s and also R"s, tbelow are $dbinom72$ methods to decide wbelow the N"s will certainly go. Similarly, tright here are now $dbinom52$ means to decide wright here the E"s will certainly go. That leaves $3$ gaps, and $3$ singleton letters, which have the right to be arranged in $3!$ means, for a total of $$inom112inom92inom72inom523!.$$


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

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


In just how many kind of means have the right to the letters of the word 'arrange' be arranged if the 2 r's and also the 2 a's carry out not happen together?
In exactly how many type of methods have the right to the letters of word $PERMUTATIONS$ be arranged if tbelow are always 4 letters in between P and S?
website style / logo design © 2021 Stack Exreadjust Inc; user contributions licensed under cc by-sa. rev2021.9.16.40224

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can keep cookies on your tool and also disclose indevelopment in accordance through our Cookie Policy.