OK, I do understand that lower E shouldn’t be played, but that’s the case for the E & A strings in the D chord as well.
Will lower E ruin the C chord more than an unfortunate A in the D chord?
It’s the same principle for all the examples you’re saying.
For example, the note E does belong in the C major chord, BUT the root note should be the lowest note in the chord, so it “sticks out”. The E note on the thickest string is one octave higher than the one on the 2nd fret of the 4th string, it would be more “noticeable” than the C note on the 3rd fret of the A string, and we don’t want that.
Of course this is my very basic understanding of it, my advice would be to try it out and try to notice.
For me, for the D chord I don’t really notice that much of a difference if I play the open A string, so I don’t worry too much about it.
Hopefully a more experienced player can answer your question more clearly.
Cheers!
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This may be a good opportunity to learn some very basic music theory. Consider the notes in each of these major chords (if you don’t know them you can either investigate the scale degrees on your own or you can just look them up). What scale degree is A in a D major scale? What scale degree is E in a C major scale? Now play each of these chords “incorrectly” (with the bass A in the D chord and the bass C in an E chord). Which one sounds “better”, or more true to the intended chord to your ears?
You should notice that the E in the bass doesn’t completely wreck a C chord, but it should sound a bit worse than a bass A in the D chord, and that should make sense based on the theory above as well.
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Making an E the bass note for a C and an A the bass note for a D will not sound completely awful since both are notes in the chord. But you will tend to ‘muddy’ the overall progression up by not playing the chord’s root note as the bass note. For example if playing G C D G, it will sound better if the lowest notes played are G C D G.
Later you will learn to play alternative bass notes for musical reasons, such as alternating the bass note when playing a C chord between the C on the A string and the G on the E string. But depending on where you are in your learning, this may be something that you are not yet ready for.
What is to be avoided is playing the open low E string when playing a D chord. That note is not a note from the D major chord and will sound off. So work hard when practicing strumming for a D chord to strum the thinnest 4 strings and if you happen to catch the open A every now and then it is not the end of the world. But strive to develop accuracy as you develop as a player.
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I’ve actually played for some time. I started when I was 18, and now I’m 52. 
But I’ve never learned it all from start, and I think I’ve missed important stuff like music theory and common practice. For instance I’d never heard about muting the E string, even if I do play alternating G as base in the C chord. I will learn it all now!
Justin is great, your answers to my question are also top notch. Thanks!
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And I omitted to say “Welcome to the Community”, Fredrik, I suggest you introduce yourself here - #community-hub:introduce-yourself. Helps to get a sense of background and proficiency level to offer as constructive answers as possible (not that I am in anyway an expert player or source of knowledge).
Hello @Eternity666 and welcome to the Community.
You have some good answers and information already.
I note in a further comment you mention about the alternating bass note when playing a C chord - going between C root note on the 5th string and G on the 6th string. This is very typical in certain musical styles and many teachers will reference Johnny Cash in demonstrating it.
On an A chord you can do this very same thing with the open 5th string (root note) and the open 6th string.
On a D chord you can do it with the open 4th string (root note) and the open A string.
You would not do it on a C chord with the root note and then the open 6th string however. And this is exactly what @ohmygato was pointing you towards.
C chord formula: 1, 3, 5 = notes C, E, G
A chord formula: 1, 3, 5 = notes A, C#, E
D chord formula 1, 3, 5 = notes D, F#, A
Look at the 1s and the 5s.
Those are the notes that work in the alternating bass style of chord playing.
Not the 3s.
Hope that helps.
Cheers 
| Richard_close2u | JustinGuitar Official Guide
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Lots of great advice and going into greater depth.
I can only reiterate that whilst it will never sound “wrong” with the low E it does tend to muddy the sound. It’s an ear thing. Not something to beat yourself up over if you accidentally play it, but it does sound brighter without.
As with any chord my advice is to properly listen and hear the difference. It is your ear and yours only that can decide which is better for a particular situation. Analyse why one sounds better to you.
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Here is an example of a situation when C/E (a C major chord with E in the bass) is exactly what is needed.
Note, the low C root is muted so as not to muddy the sound.
52 secs in the demonstration.
4 mins 50 secs in the lesson.
Imagine
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Hi, I’m new here.
I was surprised by this in the C Chord lesson. Since we’re supposed to be able to skip the bottom 1-2 strings for other chords, and that seems like a skill that’s do-able, why are we doing it differently for this chord?
Is it ideal to do both - skip the low E string for strumming, and the muting is in case you mess that up?
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Hi there,
If the open (low) E string is played, then E will be the lowest note of the chord, turning it into C/E.
I mean, why are we muting the string with a finger as well as (or instead of ?) just not playing it?
As I understand the beginners grade 1 has the thumb placed behind the neck to help build the muscle between your thumb and 1st finger in anticipation of learning to play the F barre chord in Grade 2. Therefore, at the grade 1 stage the low E string cannot be muted with the thumb so we have to make sure that we miss it when playing chords like the open A or open D. At the end of Grade 2 you are introduced to using the thumb to mute the low E string to reduce the chances of hitting the wrong notes.
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My experience with “just not playing it” is that the extreme precision required to hit the critical root note string solidly - while completely avoiding one right beside it - interferes with my strumming, and hence rhythm and dynamics. And especially on the C chord!
I think Justin says somewhere that there are great players who don’t mute, but it’s my impression that this is rare.
Justin teaches muting the E string with the ring finger for the C chord. Apparently my guitar has wider string spacing, because I tried to make that work for weeks and weeks, and never could. So I’m starting over with thumb muting.
Tedious, but my C, A, and D chords are starting to sound noticeably better now.
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Because C chord is named after C note which is root and needs to be the lowest one in the chord. If you play open top E string it is no longer the case and chord has lowest note E which makes it C/E as Jozsef commented on earlier. There is a lesson where Justin explains the difference. You can play it with top string not being muted but your tonal centre changes slightly and this is no longer a C chord in a classic view.
In real terms someone more experienced will pick it up, an average Joe or a beginner probably won’t hear the difference.
Well, “just not playing it” may require great precision, especially at a fast tempo.
A related practice is Keith Richards’ 5-string Telecaster n open G tuning which doesn’t have a low E string as it just got in the way for him.
Ah, thanks for the answers!
My guitar has narrow spaced strings, but I have narrow fingers, my ultra-trimmed fingernail keeps brushing the E string and sounding really bad!
I’m already having to work extra hard on good thumb position, thanks to my physiotherapist who showed me I need to be careful not to just lock and hyperextend one of the joints. So, I guess I’ll continue without finger muting, get more good thumb position practise, and I’ll add the thumb muting when I get to grade 2.
I can confirm this. I’ve a medical issue with my third finger and I didn’t pay too much attention to mute the 6th string, because I thought I might be easier just not to play it. Worked well till I started to use a strumming pattern like OF. Transition from G to C drives me crazy, as I was hitting the 6th string more often than not while playing the first strum after changing. It took me weeks or month with a lot of repetitive practice to improve.
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