Hello Runty Originally Posted by

**Runty** There are two parts to this one. They are listed word for word.

- How many 12 lowercase letter strings are there that contain at least 2 vowels (a vowel is one of {a,e,i,o,u})?
- How many bit strings

- of length 12 contain 8 consecutive zeros?
- are there of length at least 5 and at most 10?

Please show your work.

1. Assuming that repetition is allowed, there are $\displaystyle 26^{12}$ strings altogether. Of these, there are $\displaystyle 21^{12}$ that contain no vowels.

To construct a string containing exactly one vowel:there are $\displaystyle 12$ positions in which the vowel might be placed;

there are $\displaystyle 5$ choices of vowel;

there are then $\displaystyle 21^{11}$ choices of consonant for the remaining $\displaystyle 11$ places.

Total: $\displaystyle 12\times5\times21^{11}$

So the number of $\displaystyle 12$ letter strings that contain $\displaystyle 2$ or more vowels is:$\displaystyle 26^{12}-21^{12} - 12\times5\times21^{11}$

which is quite a lot!

2a. I am assuming this means exactly $\displaystyle 8$ consecutive zeros. So consider the 'block' containing these $\displaystyle 8$ zeros. This is to be positioned along with $\displaystyle 4$ other bits.

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

the bit immediately adjacent to the block must be a $\displaystyle 1$;

there are then $\displaystyle 2^3 = 8$ choices of the remaining 3 bits.

Total: $\displaystyle 2\times8 = 16$

If the block of $\displaystyle 8$ zeros is not at one end:there are $\displaystyle 2$ choices of position for the block;

the bit on either side of the block must be a $\displaystyle 1$;

there are then $\displaystyle 2^2=4$ choices for the remaining two bits.

Total: $\displaystyle 2\times 4 = 8$

So the overall total is $\displaystyle 16+8 = 24$.

2b. There are $\displaystyle 2^n$ bit strings of length $\displaystyle n$. So there are:

$\displaystyle \sum_{n=5}^{10}2^n$

bits strings of length $\displaystyle 5$ to $\displaystyle 10$ inclusive.

I reckon that adds up to $\displaystyle 2016$.

Grandad