How many bitstrings of length 10 have the following property:
a) sum of first $5$ bits are $3$?
b) sum of first $5$ bits equals sum of last $5$ bits?
c) the bits are written in increasing order (no $0$ after $1$)?
d) first and last bits are identical?
I guess the answer on a) is $C_5^3 \cdot 2^5 $ because we get 5 arbitrary positions and we need 3 $1$'s at the start and the answer on d) is $2^9$ because we have 8 arbitrary positions $2$ times for $0$ then $1$.
If those are correct, how to approach b) and c) ?