Closed 6 year ago.
You are watching: How many ways to arrange letters in a word
Using all the letter of the word plan how numerous different words making use of all letters at a time can be made such the both A, both E, both R both N take place together .
$egingroup$ In general if you have $n$ objects through $r_1$ objects that one kind, $r_2$ objects of another,...,and $r_k$ objects the the $k$th kind, they have the right to be i ordered it in $$fracn!(r_1!)(r_2!)dots(r_k!)$$ ways. $endgroup$
"ARRANGEMENT" is an eleven-letter word.
If there were no repeating letters, the price would simply be $11!=39916800$.
See more: 8+8=0 Hence, The Sum Of Deviations About The Mean Always Equals What?
However, due to the fact that there room repeating letters, we need to divide to eliminate the duplicates accordingly.There room 2 As, 2 Rs, 2 Ns, 2 Es
Therefore, there space $frac11!2!cdot2!cdot2!cdot2!=2494800$ methods of arranging it.
The word setup has $11$ letters, not all of them distinct. Imagine the they are written on small Scrabble squares. And also suppose we have $11$ continually slots right into which to put these squares.
There room $dbinom112$ means to choose the slots wherein the two A"s will go. Because that each of these ways, there room $dbinom92$ means to decide where the two R"s will certainly go. For every decision about the A"s and R"s, there room $dbinom72$ ways to decide where the N"s will certainly go. Similarly, there are currently $dbinom52$ means to decide where the E"s will go. That pipeline $3$ gaps, and also $3$ singleton letters, which have the right to be i ordered it in $3!$ ways, for a total of $$inom112inom92inom72inom523!.$$
Highly energetic question. Earn 10 call (not counting the association bonus) in order to answer this question. The reputation necessity helps defend this question from spam and non-answer activity.
Not the prize you're spring for? Browse other questions tagged permutations or questioning your own question.
In how many ways deserve to the letters of the word 'arrange' be i ordered it if the two r's and also the 2 a's do not occur together?
In how plenty of ways deserve to the letters of indigenous $PERMUTATIONS$ be arranged if over there are always 4 letters between P and also S?
site architecture / logo © 2021 stack Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.10.29.40598