Question 1201883
.
How many 11-bit strings (that is, bit strings of length 11) are there which:
a) Start with the sub-string 011?
b) Have weight 6 (i.e., contain exactly 6 1’s) and start with the sub-string 011?
c) Either start with 011 or end with 01 (or both)?
d) Have weight 6 and either start with 011 or end with 01 (or both)?
~~~~~~~~~~~~~~~~



        Too many questions for one post. 
        I will answer first two to get you start.



<pre>
(a)  In this part, you consider 11-bit strings (the string consisting of ones and zeroes).
     that starts with substring 011.


     So, the remaining part of these strings is of the length 11-3 = 8.


     In these 8 positions, we may have 0 or 1 in each position, 
     independently of what is in other 7 remaining positions.


     The number of possibilities is  {{{2^8}}} = 256.    <U>ANSWER</U>



(b)  In this part, you consider 11-bit strings, that have precisely 6 ones
     and start with the substring 011.


     So, the remaining part of these strings is of the length 11-3 = 8 
     and has precisely 6 ones.


     The number of ways to select 6 positions among 8 positios for these 6 ones
     is  {{{C[8]^6}}} = {{{8!/(6!*2!)}}} = {{{(8*7)/(1*2)}}} = 4*7 = 28.    <U>ANSWER</U>
</pre>

Solved.


-----------------


Regarding two other parts &nbsp;(c) &nbsp;and &nbsp;(d), &nbsp;keep in mind that in my post
I gave you &nbsp;ALL &nbsp;NECESSARY &nbsp;ideas on how to complete the remaining parts.


So, &nbsp;if you really want to learn the subject, &nbsp;you should apply this knowledge
and complete parts &nbsp;(c) &nbsp;and &nbsp;(d) &nbsp;ON &nbsp;YOUR &nbsp;OWN.