# Tagged Questions

For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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### Closed formula for ${r \choose 1}+{r \choose 2}\cdots{r \choose w}$ where $w < r$ [closed]

Let $r,w \in \mathbb{N}$. Are there some formula for the next sum? $${r \choose 1}+{r \choose 2}\cdots{r \choose w}$$ where $w<r$?
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### Combinatorial identity's algebraic proof without induction. [duplicate]

How would you prove this combinatorial idenetity algebraically without induction? $$\sum_{k=0}^n { x+k \choose k} = { x+n+1\choose n }$$ Thanks.
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### Formation of Teams in Permutation and Combination

A class has $n$ students , we have to form a team of the students including at least two and also excluding at least two students. The number of ways of forming the team is My Approach : To include ...
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### Counting problem: How many triangles?

Sixteen points are on the sides of a $4\times 4$ grid so that the center portion of $2\times 2$ are removed. How many triangles are there in total that have vertices chosen from those remaining points ...
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### Colors on sets $S=\{1,2 \cdots ,1000\}$.

To each element of sets $S=\{1,2 \cdots ,1000\}$ a color is assigned. Suppose that for any two elements $a$ and $b$, of $S$,if $15$ divides $a+b$, then they both are assigned with same color. What is ...
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### How many 3 digit numbers that the sum of their digits equals 12?

How many positive 3-digit numbers exist such that the sum if their digits equals 12? A) 54 B) 61 C) 64 D) 65 E) 66 I believe the answer is E. Online problems state that is a stars and bars ...
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### Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
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### A 3-valued mathematical logic?

Classical propositional logic is consistent and in conformity with human language. A formal statement is true or not true and it is possible to develope rules with which it is possible decide which ...
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### Probability question involving sets of cards

I have an infinite deck built out of sets of 10 cards (in other words 10*n cards). The sets are identical so one '2' is identical to another '2'. A player draws 6 cards. If he draws: any '1' AND a '...
the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was ...