Hello Runty1. Assuming that repetition is allowed, there are strings altogether. Of these, there are that contain no vowels.

To construct a string containing exactly one vowel:there are positions in which the vowel might be placed;Total:

there are choices of vowel;

there are then choices of consonant for the remaining places.

So the number of letter strings that contain or more vowels is:which is quite a lot!

2a. I am assuming this means exactly consecutive zeros. So consider the 'block' containing these zeros. This is to be positioned along with other bits.

If the block is at one end of the bits:there are two choices of end;Total:

the bit immediately adjacent to the block must be a ;

there are then choices of the remaining 3 bits.

If the block of zeros is not at one end:there are choices of position for the block;Total:

the bit on either side of the block must be a ;

there are then choices for the remaining two bits.

So the overall total is .

2b. There are bit strings of length . So there are:

bits strings of length to inclusive.

I reckon that adds up to .

Grandad