# Tag Info

### How many ways can $7$ professors and $5$ students be seated at this long rectangular table so that no student sits across from another student?

The problem with your solution is that you're not accounting for the possibility that more than one pair of students sits across from one another, so you're double counting those combinations. You ...
• 17.1k

### How many arrangements we can make from the word "SINGAPORE" where the letters $E$ and $I$ do not occur together?

You can take all the possibilities and subtract the cases where the letters $E$ and $I$ occur together. Hence the desired result can be expressed as $9! - 2\times 8! = 282240$. Thus your answer is ...
• 7,791
Accepted

• 25.5k
1 vote
Accepted

### Evaluating the series $\sum_{i=0}^{\infty} \sum_{j=0}^i \sum_{h=0}^j \frac{a^h(2a)^{j-h}(3a)^{i-j}}{(i+3)!}$

It is just an iterated Cauchy product of the absolutely convergent series $$\sum_{n=0}^{+\infty} \frac{a^n}{(n+1)!} = \frac{e^a-1}{a}.$$ by itself. We get \sum_{n=0}^{+\infty} \Big(\sum_{p+q+r=n} \...
• 2,850

Only top scored, non community-wiki answers of a minimum length are eligible