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What is the number of strings of length $235$ which can be made from the letters A, B, and C, such that the number of A's is always odd, the number of B's is greater than $10$ and less than $45$ and the Number of C's is always even?

What I can think of is

$$\left(\binom{235}{235} - \left\lfloor235 - \frac{235}2\right\rfloor\right) \binom{235}{35} \binom {235}{ \lfloor 235/2\rfloor}\;.$$


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The formula you have written down is going to give you a negative number, so there is probably room for improvement. – Gerry Myerson Feb 23 '12 at 6:19
up vote 1 down vote accepted

If there is an odd number of A's and an even number of C's, it follows that there must be an even number of B's.

Therefore, to construct a string of length 235, satisfying the properties above, it is sufficient to specify the position of the A's and the B's, because then the C's will be forced.

There are ${235 \choose n}$ ways to specify the position of the B's, where $n\in\{12,14,\dots,44\}$.

After we have placed the B's, there are $235-n\choose m$ ways to place the A's, where $m \in \{1,3,5,\dots,\frac{(235-n-1)}{2}\}$.

Hence, the number of possible strings is:

$$\displaystyle\sum_{n\in\{12,14,\dots,44\}}_{m\in\{1,3,\dots,(235-2n-1)/{2}\}} {235 \choose n} {235-n\choose m}$$

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It's the coefficient of $x^{235}$ in $$(x+x^3+x^5+\cdots)(x^{11}+x^{12}+\cdots+x^{44})(1+x^2+x^4+\cdots)$$ Use the formula for sum of a geometric progression, then use the binomial theorem, it should fall right out.

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Hint: I would do it with a program. You have 44*2*2 allowable states-number of B's, parity of A's and parity of C's. Write the recurrence relations and note that the empty string has even C's and no B's.

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