phbern
If I am understanding you correctly that a person can choose form 1-10. You can do something like this where you need to keep the videos in separate blocks and your question in a separate block.

Hope this helps!

You can do Question Randomization (this option is available when you select the block) and using display logic in the question level to achieve the setup.

Thanks Deepak and PraDeep for your suggestions. I think, though, that I wasn't entirely clear about my question.
The respondent can select any combination of Choice 1, Choice 2, Choice 3 and Choice 4. What they select determines which videos will make up the "population" from which only one will be randomly shown to the respondent. So...

Respondent 1 selects Choice 1, Choice 2 and Choice 3; will be randomly shown ONE of video 1, video 2 OR video 3
Respondent 2 selects Choice 2 and Choice 4; will be randomly shown ONE of video 2 OR video 4
Respondent 3 selects Choice 1; will be shown video 1

Deepak, I had an idea similar to yours, but the glitch is we don't want to show all videos. we only want to show one, and which one is based on selections made by the respondent.

You repeated the choices, so it doesn't make sense. Instead, let's say for choices 1 or 2 you want to randomize video 1 or 2, for choice 3 or 4 you want randomize video 3 or 4, and for choice 5 you want to show video 5. Use survey flow branching and randomizer to set the video url. Then pipe the url into your 'video' question:
If Choice 1 or 2
Randomizer: 1 of 2 evenly distributed
Set Embedded Data: videoUrl = {url of video 1 goes here}
Set Embedded Data: videoUrl = {url of video 2 goes here}
If Choice 3 or 4
Randomizer: 1 of 2 evenly distributed
Set Embedded Data: videoUrl = {url of video 3 goes here}
Set Embedded Data: videoUrl = {url of video 4 goes here}
If Choice 5
Set Embedded Data: videoUrl = {url of video 5 goes here}
Video Block

https://community.qualtrics.com/XMcommunity/discussion/comment/50509#Comment_50509In this case, if you want the selection to determine which videos are possible, but randomly present the possible videos, I think what you want to do is "nest" branches inside your randomizer, like this:
One video block will play (per the randomizer) but it will be selected based on the selection of the question.
If in fact what you have is (going back to the original post) a need to TERMINATE if they select ONLY 1,2,3,8,9,10, but if they select ANY of 4,5,6,7 then present one of the videos that match their selection that might look like this
The first logic is checking that they didn't select ANY of the desired answers (4-7) and terminating if not. Then for those who did select one of the desired answers (4-7) the nested logic in the survey flow will randomly present one of the videos, based on how they answered (eg only selected #4 = only show video#1; selected 4 & 6 = randomly show video #1 (for selection4) or video #3 (for selection6).
I think this may close to what you are envisioning?

Thank you, everyone for your help!
TomG, your solution is the one that will work. There are 15 possible combinations, so setting it up will be a bit tedious - but such is life!

Thank you everyone Deepak PraDeepKotian_Ugam CarolK for contributing to this tricky question! Great solution by TomG and we truly appreciate you all going above and beyond to help our community!