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I was solving a problem in probability. Here is the statement.

Seven terminals in an on-line system are attached to a communications line to the central computer. Exactly four of these terminals are ready to transmit a message. Assume that each terminal is equally likely to be in the ready state.Let ##X## be the random variable whose value is the number of terminals polled until the first ready terminal is located.

(a) What values may ##X## assume ?

(b) What is the probability that ##X## will assume each of these values? Assume that terminals are polled in a fixed sequence without repetition.

(c) Suppose the communication line has ##m## terminals attached of which ##n## are ready to transmit. Show that ##X## can assume only the values ##i=1,2,\cdots,m-n+1## with ##P[X =i] = \binom{m-i}[n-i}/\binom{m}{n}##. The problem is from the book Probability, Statistics, and Queueing Theory: With Computer Science Applications by Arnold Allen. The problem can be seen in this google book review. Now I have been able to solve this problem. Though its not asked here, wanted to find the ##E(X)## for case (c). So we would have $$\sum_{i=1}^{m-n+1}i \frac{\binom{m-i}[n-i}} {\binom{m}{n}}$$

I have not been able to get this into some nice closed form. Can anybody provide guidance ? Also there seems to be some problem with the latex command \binom{}{}. Its not working for me.

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# I Need closed form for a Binomial series

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