Let $X$ be an integer valued random variable such that $$E\left[X(X-1)\ldots(X-k+1)\right]=\begin{cases}\binom{n}{k}k!&,\text{ if } k=0,1,2,\ldots,n \\ 0&,\text{ if }k>n\end{cases}$$, then prove that $X$ can be degenerate at $n$.
What I am trying to do is that to use probability generating function but not able to do so.
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You are given the $k$th order factorial moment of $X$, denoted by $$E\,[(X)_k]=E\left[X(X-1)\ldots (X-k+1)\right]$$
Find the factorial moment generating function $M(t)$ of $X$ explicitly, i.e., by expanding $$M(t)=\sum_{k=0}^{\infty} \frac{t^k}{k!}E\,[(X)_k]$$
Keeping in mind that $M(t)$ can also be expressed as $$M(t)=\sum_{k=0}^{\infty} \frac{t^k}{k!}\sum_{j=0}^n (j)_kP(X=j)$$
, equate coefficients in both expressions to find the exact pmf $P(X=j)$.
Hint: Take expectation to $$ (X-n)^2=X(X-1)-(2n-1)X+n^2. $$