Probability on dispersing of reports

A little aid below. Workout 21, Ch. 2 from Feller's book reviews

In a community a $n+1$ citizens, an individual informs a report to a 2nd individual, that subsequently repeats it to a 3rd individual, etc At each action, the recipient of the report is picked randomly from the $n$ individuals readily available. Locate the probability that the report will certainly be informed $r$ times without : a) going back to the mastermind, b) being duplicated to anybody. Do the very same trouble when at each action the report is informed by someone to a celebration of $N$ arbitrarily picked individuals. (The first inquiry is the grandfather clause N = 1).

I currently did a) and also b) for the first summary of the trouble and also a) for the instance when the report is spreading out via a celebration of $N$ individuals, nonetheless, my remedy for b) in this 2nd instance is not deal with.

I reasoned in the list below means : In a first instance, $n$ individuals to receive the report, nonetheless, it is required to spread out such report via a team of $N$ individuals, consequently, there are $\displaystyle n \choose N$ means to pick those celebrations. As soon as among these individuals is picked, he/she can pick from an additional celebration of $N$ individuals, caring for passing by a person that currently recognize the report, which is, there are $\displaystyle n-1 \choose N$, and more, till we get to the $r$ action in this procedure. Consequently, the probability I get is :

$$\frac{\displaystyle {n \choose N} {n-1 \choose N} {n-2 \choose N} ... {n-r+1 \choose N}}{\displaystyle {n \choose N}^{r}}$$

According to guide, the remedy has to be $\displaystyle \frac{(n)_{Nr}}{(n_{N})^{r}}$ (which is not equal to the first expression).

I will certainly value any kind of aid.

2019-12-02 02:49:51
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Answers: 1

Liberalkid is right. Utilizing his pointer, you get $$\frac{\binom{n}{N}\binom{n-N}{N}\cdots\binom{n-(r-1)N}{N}}{\binom{n}{N}^r} = \frac{n_N (n-N)_N \cdots (n-(r-1)N)_N}{(n_N)^r} = \frac{n_{Nr}}{(n_N)^r}.$$ In the first action you terminate $N!$ from each side $r$ times.

2019-12-03 01:10:53