Answer: c. 1 divided by 204.
Step-by-step explanation:
Given: The number of friends = 4
The total number of sweatshirts =18
If each one orders a different color (no repeats) at random., then the total number of ways to order sweatshirts =![^{18}P_4=\frac{18!}{(18-4)!}=\frac{18\times17\times16\times15\times14!}{14!}=73440](https://tex.z-dn.net/?f=%5E%7B18%7DP_4%3D%5Cfrac%7B18%21%7D%7B%2818-4%29%21%7D%3D%5Cfrac%7B18%5Ctimes17%5Ctimes16%5Ctimes15%5Ctimes14%21%7D%7B14%21%7D%3D73440)
If the friends order only cotton sweatshirts, then the number of ways to order sweatshirts=![^6P_4=\frac{6!}{(6-4)!}=\frac{6\times5\times4\times3\times2!}{2!}=360](https://tex.z-dn.net/?f=%5E6P_4%3D%5Cfrac%7B6%21%7D%7B%286-4%29%21%7D%3D%5Cfrac%7B6%5Ctimes5%5Ctimes4%5Ctimes3%5Ctimes2%21%7D%7B2%21%7D%3D360)
Now, the probability that the friends order only cotton sweatshirts =![\frac{360}{73440}=\frac{1}{204}](https://tex.z-dn.net/?f=%5Cfrac%7B360%7D%7B73440%7D%3D%5Cfrac%7B1%7D%7B204%7D)
Hence, the probability that the friends order only cotton sweatshirts =![\frac{1}{204}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B204%7D)