Question

Find the number of permutations that can be formed from all the letters of word ELEVEN.

Answer 1

Hi friend

. I'm going to solve the problem for the word SEVENTEEN and you can do it for ELEVEN.

First, I count up all the letters: I have 9 letters.

Then I look for duplicates: 4 E's and 2 N's.

Now I use the "permutations with duplicates" formula, which says

where n is the total number of letters and ri, rii, etc are the repeating counts. So in this case I have two repeating letters, so ri = 4 and rii = 2. So the answer to the first part of your question is

For the second part of your question (how many begin and end with an E), I just remove the two E's that have to go at the beginning and the end and I don't think about them anymore. So now I do the same calculation for SVNTEEN: 7 letters with 2 E's and 2 N's:

For the third part of your question, I just group three of the E's together and think of it as a single "letter". So the "letters" I have are: "EEE",S,V,N,T,E,N: 7 letters with 2 N's. Use the same equation:

And for the last part of the question: "How many begin with E and end with N?", I remove an E and an N from and ignore them because they each have only one place to go. So left over I have SVETEEN: 7 letters, with 3 E's: