Take the following steps to enable the "shuffle Mode" on your YouTube "Playlist."

1. Launch YouTube,

2. At the bottom of the phone are five tabs--Starting from the bottom left, going right, you've got:
"Home," "Trending," "Subscription," "Activity" and "Library."

3. Tap on "Library,"

4. On the screen that shows up upon tapping on "Library," find and tap on the name of the "Playlist," in this case, for this demonstration, I have created one called "Mix Bag."

5. After tapping on the "Playlist" in question, find and tap on "Play all button," found on the right edge of the phone, just two or so inches from the top right, just below the "playlist" title.

6. Now that you've tap on "Play all button," all the videos or songs in this "Playlist" are displayed,

7. Next, starting from the very top-left corner, explore downwards to find "Shuffle button" which is just below the title of the "Playlist,"

8. To the right of the "shuffle button" is the "repeat button." We notice that the "shuffle button" simply says: "Shuffle button,"

9. Tapping on the "shuffle button" turns the "shuffle mode" on, and if we move our finger away from that button after tapping on it to the right, which would be the "repeat button" and move our finger back to the "shuffle button," this time, we will hear "Selected, shuffle button." Meaning, the shuffling of the playlist has been activated.

10. This button, is a toggle button, so if you think you don't want the music shuffled, then simply tap on the button that says: "Selected, shuffle button" and doing so takes off the shuffling mode, and now that button simply reads: "Shuffle button" instead of "Selected, shuffle button."

That's it!
You've enabled the "shuffle mode" on your "playlist" in the YouTube app.