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)?
Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website! .
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)?
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Too many questions for one post.
I will answer first two to get you start.
(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  = 256. ANSWER
(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  =  =  = 4*7 = 28. ANSWER
Solved.
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Regarding two other parts (c) and (d), keep in mind that in my post
I gave you ALL NECESSARY ideas on how to complete the remaining parts.
So, if you really want to learn the subject, you should apply this knowledge
and complete parts (c) and (d) ON YOUR OWN.
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