You are watching: How many permutations of 4 letters
Consider the multiset $M=<1 extE, 1 extX, 2 extA, 1 extM, 2 extI, 2 extN, 1 extT, 1 extO>$.
The number of $4$-permutations of $M$ is the coefficient of $x^4/4!$ in the exponential expansion of
$$(1+x)^5 left(1+x+dfracx^22! ight)^3.$$
The coefficient of $x^4$ above is $409/4$. Multiplying by $4!$ we get $2454$.
The $(1+x)$ represent the letters that are not repeated, while $(1+x+x^2/2)$ represents those letters that are repeated twice.
The word examination has $11$ letters. It is composed of $8$ different letters, $3$ of which repeat once.
As Andre suggests, break it down into three cases.All different. Choose the four letters out of the eight you"ll use $8 choose 4$, and then order them ($4!$).One double. Choose which of the three letters is the double $3 choose 1$, then choose which positions they go in the word $4 choose 2$. Then, choose the two remaining letters $7 choose 2$, and decide which goes first and which goes second in the remaining two spaces ($2!$).Two doubles. Choose which two of the three doubled letters to use $3 choose 2$, then order them ($6$).
Adding all of these gives $2454$.
Thanks for contributing an answer to les-grizzlys-catalans.orgematics Stack Exchange!Please be sure to answer the question. Provide details and share your research!
But avoid …Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.
Use les-grizzlys-catalans.orgJax to format equations. les-grizzlys-catalans.orgJax reference.
See more: How Can I D Is Zinc Chloride Soluble In Water ? Zinc Chloride
To learn more, see our tips on writing great answers.
Post Your Answer Discard
Not the answer you're looking for? Browse other questions tagged combinatorics permutations or ask your own question.
How many selections of three cards can be made from cards bearing the letters of the word EXAMINATION if ...?
site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.11.16.40760