5 votes

Let $x_i \in \mathbb Z,$ such that $|x_1|+|x_2|+|x_3|+...+|x_{10}|=100.$ Find number of solution.

Let $r$ be the number of zeros in $x_1,x_2,\cdots, x_{10}$ where $0\le r\le 9$. There are $\binom{10}{r}$ ways to choose $r$ zeros in $x_1,x_2,\cdots,x_{10}$. The number of non-zero $x_i$ is $10-r$. ...
mathlove's user avatar
  • 141k
3 votes

Simple solution to random walk

One way to solve this problem is to represent the possible courses of the match by paths through a directed graph. Each vertex of the graph represents a possible score $\ (h,a)\ $ that might have ...
lonza leggiera's user avatar
3 votes

Simple solution to random walk

With the $X$-axis representing score of the home team and the $Y$-axis that of the away team. write down scores as $0-3\mid 1-3 \mid 2-3 \mid 3-3 \mid 4-3$ $0-2\mid 1-2 \mid 2-2 \mid 3-2 \mid 4-2$ $0-...
true blue anil's user avatar
1 vote

Number of ways to arrange characters in the alphabet

$T$ is the number of the natural solutions of the equation $x_1+\cdots+x_A=S$, where $x_i$ is the number of $i$th character in the soup, given by $$T=\binom{S-1}{A-1}={{(S-1)!}\over{(A-1)!(S-A)!}}.$$ ...
Amir's user avatar
  • 5,164
1 vote

Correctness of Solution for Forming a Committee with More Democrats than Republicans

To elaborate on the discussion in the comments: The given method is flawed as it multiply counts any committee that has more than the minimal number of Democrats. To avoid that, we split into even and ...
lulu's user avatar
  • 70.5k
1 vote

combinations problem - teachers to groups

This is a bizarre sentence to read! I tried Benjamin Dickman's formula but it did not work out =, the right answer is 15504 I don't even know where to start. I will guess that you are referring to ...
Benjamin Dickman's user avatar

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