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

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?

Answer 1

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 $1260-360-360+120=660$