This is a variation of question asked on this site before.  Consider a set with $𝑎_1$ 'distinct' 1s, $𝑎_2$ 'distinct' 2s, ... , $𝑎_𝑛$ 'distinct' ns. You have $𝑎_1+1$ choices for the 1s (including the option of none of them being chosen) and similarly for the other elements. The total number of subsets is therefore  $(𝑎_1+1)(𝑎_2+2)...(𝑎_𝑛+1)$  Now how to find the number of subsets with size $\leq $ 'k'.
The problem of counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $a_i\geq0$ for $i\in\{1,2,\ldots,n\}$ such that $$a_1+a_2+a_3+....a_n=N$$ can be solved with a stars-and-bars argument. What is the solution if one adds the constraint that $a_i\leq r_i$ for certain integers $r_1,\ldots,r_n$?  e.g. for $n=3$, $N=6$ and $(r_1,r_2,r_3)=(3,3,2)$, the tuple $(a_1,a_2,a_3)=(2,3,1)$ is a solution, but $(2,1,3)$ is not a solution because $a_3=3&gt;2=r_3$.
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I know what I must do (for a $v$ in the range of $T^*$, I have to show that $v\perp w$ for every $w$ in $\ker(T)$ and then do the opposite), but I don't know how to show that this inner product is zero. I have a SQL script that creates a package with a comment containing an ampersand (&amp;). When I run the script from SQL Plus, I am prompted to enter a substitute value for the string starting with &amp;. How do I disable this feature so that SQL Plus ignores the ampersand? Functional equations leading to sine and cosine There are $2n+1$ people. For each $n$ people there is somebody who is friend with each of them. Prove there is a "know-them-all" person. I have a following problem.     There are $2n+1$ people in the room. For any group of $n$ people there is a person (different from them) who is friends with each person in this group. Prove that there is a person, who knows all $2n$ other people.   One can easily see, that $\min_v \deg v = n$. 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When you insert an image (or table etc.) caption in Word, you get  dialog. You enter your caption,  "A happy bicycle" (in yellow).  This might be fine for a happy bicycle, but I want to have captions that have one style for the "enumeration" (label and number) and one style for the description (caption text, as called by Word), as such:    I have tried searching for this but I haven't found anything yet...