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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) ?

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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:

$$\sum_{i=0}^{10}1 = 11$$

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