How many $11-$bit strings $($that is$,$ bit strings of length $11)$ are there which:

a$.$ Start with the sub$-$string $011?$

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b$.$ Have weight $8($i$.$e$.,$ contain exactly $81\text{'}$s$)$ and start with the sub$-$string $011?$

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c$.$ Either start with $011$ or end with $01($or both$)?$

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d$.$ Have weight $8$ and either start with $011$ or end with $01($or both$)?$

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