<s>[INST] $6$ women and $4$ men wait in line. If their order in line is random, find the probability that all of the women are adjacent to one another. My thoughts on the problem are that the number of ways the women can be adjacent to each other is $5!$ and the total number of arrangements for all the people is $10!$. Is this correct?
 [/INST] Total number of arrangements was correctly determined as $10!$.
The number of ways the $6$ women can all stand next to each other are easily seen through the following:
$W_{1} W_{2} W_{3} W_{4} W_{5} W_{6}$ __ __ __ __
__ $W_{1} W_{2} W_{3} W_{4} W_{5} W_{6}$ __ __ __ 
__ __ $W_{1} W_{2} W_{3} W_{4} W_{5} W_{6}$ __ __
__ __ __ $W_{1} W_{2} W_{3} W_{4} W_{5} W_{6}$ __
__ __ __ __$W_{1} W_{2} W_{3} W_{4} W_{5} W_{6}$
So, we have $5$ ways for the women to be next to each other and $4!$ ways to arrange the men. Next, we need to determine how many different ways the women can be arranged, which is $6!$
Thus, the total number of ways the women can be next to each other is: $(5\cdot 4!)(6!)$
Probability all women are adjacent: $\large\frac{5!6!}{10!}$
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