We would like to count how many ways 3 boys and 3 girls can sit in a row. How many ways can this be done if:

(b) all the girls sit together? Since all the girls must sit together, we treat the girls as a single unit. Then we have 4 people to arrange with 3! positions for 3 girls for a total of 4!3! ways to arrange them.

  • 1
    $\begingroup$ You’re forgetting that the girls can sit in $3!$ different orders within their block. $\endgroup$ – Brian M. Scott Jan 27 '13 at 0:53
  • $\begingroup$ The $4$ is what gave rise to your $4!$. $\endgroup$ – André Nicolas Jan 27 '13 at 0:56
  • $\begingroup$ thanks brian, i got it. 3! is the different ways the girls can sit next to each other since they are all different individuals $\endgroup$ – user59795 Jan 27 '13 at 0:57

Another way is to observe, as you did, that there are $4$ legal arrangements of the letters $b$ and $g$. For each of these arrangements, the boys can be placed in $3!$ ways, and for each of these placements, the girls can be arranged in $3!$ ways, for a total of $(4)(3!)(3!)$.

That way of thinking about things might be useful if instead we want, for example, the number of arrangements that have a girl at each end.


There are $4!$ ways to arrange the four blocks, where each boy is one block, and the three girls together are the fourth block. Once you’ve done that, you can arrange the $3$ girls in $3!$ different orders within their block. Thus, for each arrangement of the blocks you get $3!$ arrangements of the people, for a total of $4!3!$ arrangements of the people.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.