Question

In how many ways can six 3's and four 2's be arranged in a row so that the 2's are always apart?

Answer 1

\[\Large \_3\_3\_3\_3\_3\_3\_\]There are seven spaces and we have four 2's. Pick any 4 spaces out of the 7 spaces and fill them with 2's. So for example, if we picked space #2, #4, #6, #7
\[\Large \_323\_323\_3232 = 3233233232\]In how many ways can we select 4 spaces out of 7?
\[\large \left(\begin{matrix}7 \\ 4\end{matrix}\right) = \frac{ 7! }{ 4!*3! }=\frac{ 7.6.5 }{ 3.2}= 35\].

Answer 2

Thank you!

Answer 3

You are welcome.