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Number of ways of choosing (n1,n2,…,nk)

Question : There are n1,n2,…,nk objects. The order doesn’t matter. The object can be taken from any of them, any number of times. How many ways are there to pick (N1,N2,…,NK)?
My attempt : If we have k objects, then there are C(nk,k) ways to pick k from nk, C(nk,k) ways to pick k from nk+1,…,C(nk,k) ways to pick k from nk+k-1…C(nk,k) ways to pick k from nk+k-1…C(n1,k) ways to pick k from n1, C(n1,k) ways to pick k from n2,…,C(n1,k) ways to pick k from nk+k-1…
In short, the C(n1,k) ways to pick k from n1, C(n2,k) ways to pick k from n2, C(n3,k) ways to pick k from n3,… are summed up so that
there are total C(n1,k) + C(n2,k) + C(n3,k) +…. + C(nk,k) ways to pick k objects
Can someone tell me how to solve this?

A:

You were almost there, just replace sum with sum over all $k=1$ to $k=n$ and we get
$$\sum_{k=1}^n\binom{n}{k}=\frac{1}{n!}\sum_{k=1}^n n!=n!\;.$$

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