Question 1175303
.
How many 12-bit strings (that is, bit strings of length 12) are there which:
have weight 7 and either start with 011 or end with 01 (or both)?

There were three questions before this which I have worked the answers to, but I'm confused for this part.
Other questions:
a. Q. Start with the sub-string 001? A. 2^9
b. Q. Have weight 7 (i.e., contain exactly 7 1's) and start with the sub-string 001? A. C(9,5)
c. Q. Either start with 011 or end with 01 (or both)? A. 2^9 + 2^10 - 2^7
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Let me rewrite the problem (the first two lines) in a way understandable for ANYONE.


<pre>
    The word of 12 symbols is written using symbols "1" and "0".

    How many such words do exist, having 7 symbols "1" and 

        (a)  starting from "011" ?

        (b)  ending with "01" ?

        (c)  starting from "011"  and  ending with "01".

     Questions (a), (b) and (c) are three SEPARATE questions.
</pre>


Now everything is clear.  See my solution below.



<U>Solution</U>


<pre>
(a)  Three leftmost positions are just occupied and, THEREFORE, are not the subject of consideration.

     For the rest 12-3 = 9 positions, we have 7-2 = 5 "ones" to distribute.

     We can choose these 5 positions for "ones" from 9 positions by  {{{C[9]^5}}}  ways.



(b)  Two rightmost positions are just occupied and, THEREFORE, are not the subject of consideration.

     For the rest 12-2 = 10 positions, we have 7-1 = 6 "ones" to distribute.

     We can choose these 6 positions for "ones" from 10 positions by  {{{C[10]^6}}}  = {{{C[10]^4}}} ways.



(c)  Three leftmost positions and two rightmost positions are just occupied and, THEREFORE, are not the subject of consideration.

     For the rest 12-3-2 = 7 positions, we have 7-2-1 = 4 "ones" to distribute.

     We can choose these 4 positions for "ones" from 7 positions by  {{{C[7]^4}}}  ways.
</pre>

Solved.  &nbsp;&nbsp;All questions  &nbsp;(a), &nbsp;(b) &nbsp;and &nbsp;(c)  &nbsp;are answered.


Is everything clear to you from my post ?


If you still have questions, &nbsp;then let me know.



/\/\/\/\/\/\/\/



It is how a Math problem should be presented and solved.


Crystally clear for EVERYONE.



How it was worded and presented in your original post, is good for recycling, only.


Probably &nbsp;(I am ready to admit it) &nbsp;it is not your fault, &nbsp;but then it is the fault of the person, who composed this problem . . . 


Also, for the future, if you have several questions, it is ALWAYS better to separate them and to submit them ONE per POST.