# How many bitstrings of length 10 have the following property:

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) ?

Your solutions to the first and last question are correct. To solve the second question, we can divide the string in two parts. The number of ways to arrive at a sum of $$n$$ for each part equals $${5 \choose n}$$, and the number of possible strings thus equals:
$$\sum_{i=0}^{5} {5 \choose i} {5 \choose i} = 1 \cdot 1 + 5 \cdot 5 + 10 \cdot 10 + 10 \cdot 10 + 5 \cdot 5 + 1 \cdot 1 = 252$$
To solve the third question, we only have to choose the number of $$0$$s, since the remaining $$1$$s are then put at the end of the string. The number of possible strings thus equals:
$$\sum_{i=0}^{10}1 = 11$$