In how many ways can the letters of the word 'arrange' be arranged if the two r's and the two a's do not occur together?

Question

Hi and welcome to the site! Since this is a site that encourages learning, you will get much more help if you show us what you have already done. Could you edit your question with your thoughts and ideas? — 5xum, Sep 17 '14 at 07:59
This is not a valid arrangement: aangrre. But what about aanrgre, is this valid or invalid? In other words, do you want to exclude only arrangements where both letters occur together, or also arrangements where either one of the letters occurs together? — barak manos, Sep 17 '14 at 08:20

score 3 · Answer 1 · edited Jun 15 '17 at 11:01

Total number of combinations: $\dbinom{7}{2}\cdot\dbinom{5}{2}\cdot\dbinom{3}{1}\cdot\dbinom{2}{1}\cdot\dbinom{1}{1}=1260$

Number of combinations with aa: $\dbinom{6}{2}\cdot\dbinom{4}{1}\cdot\dbinom{3}{1}\cdot\dbinom{2}{1}\cdot\dbinom{1}{1}=360$

Number of combinations with rr: $\dbinom{6}{2}\cdot\dbinom{4}{1}\cdot\dbinom{3}{1}=360$

Number of combinations with aa and rr: $\dbinom{5}{1}\cdot\dbinom{4}{1}\cdot\dbinom{3}{1}\cdot\dbinom{2}{1}\cdot\dbinom{1}{1}=120$

So the number of combinations without aa or rr is $1260-360-360+120=660$

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