Based on the piece of Bach - Prelude in C (BWV 846).Why do those B- flats at the end not take us away from the Key of C major ( why arent they part of a modulation to F major)?

(A) No perfect cadence in the key of F
(B) F major chord is comprimised of F,A and C and B flat isnt one of them
(C) Not enough F's
(D) The B naturals contradict the B flats
(E) Because of the C bass note running all the way through these last bars

Definitely B

To understand why the B-flats at the end of the piece do not take us away from the key of C major, we can analyze the musical elements present in those sections.

(A) The absence of a perfect cadence in the key of F is not a determining factor in this case. While a perfect cadence would indeed help establish a sense of tonality in F major, the lack of one doesn't prevent modulation outright.

(B) The fact that the F major chord is comprised of F, A, and C, and doesn't include B-flat, is also not the reason. In music, chords can be altered or expanded beyond their basic triadic form, and the inclusion of a B-flat doesn't necessarily eliminate the possibility of modulation.

(C) The number of F's present is not a determining factor either. The number of occurrences of a specific note does not dictate the modulation's possibility.

(D) The presence of B naturals contradicting the B-flats could potentially suggest a modulation, but it alone is not sufficient evidence to conclude a key change. It merely introduces fleeting moments of harmonic tension.

(E) The C bass note running all the way through the last bars is, indeed, the crucial reason why the B-flats do not result in a modulation to F major. The continuous C bass note serves as a pedal tone or drone that anchors the tonal center of C major. Throughout these bars, the B-flats are treated as non-functional chromatic passing tones or coloristic ornaments, rather than modulating agents.

Therefore, the correct answer is (E) - the presence of the C bass note running all the way through these last bars prevents the B-flats from modulating the piece to F major.