A Very Large Set of Dominoes

Let the total number of dots of double-n dominoes be $N(n)$ , then from the arrangement shown for double-6 dominoes, we have:

$\begin{aligned} N(n) & = \sum_{j=0}^n \sum_{k=0}^j (j + k) \\ & = \sum_{j=0}^n \left(j(j+1) + \frac {j(j+1)}2\right) \\ & = \frac 32 \sum_{j=0}^n \left(j^2 + j \right) \\ & = \frac 32 \left(\frac {n(n+1)(2n+1)}6 + \frac {n(n+1)}2 \right) \\ & = \frac {3n(n+1)}{12}(2n+1 + 3) \\ & = \frac {n(n+1)(n+2)}2 & \small \color{#3D99F6} \text{Note that }N(6) = 168 \end{aligned}$

Therefore, $N(250) = \dfrac {250(251)(252)}2 = \boxed{7906500}$ .

Every for any set of double-n dominoes each number appears $n+2$ times, once paired with each number $0$ through $n$ plus one more for the double.

So the total is $0(n+2)+1(n+2)+2(n+2)+3(n+2)+...+n(n+2)$

Refactor as $(0+1+2+3+...+n)(n+2)=\frac{n(n+1)}{2}\times(n+2)$

So the formula for the number of dots is $n(n+1)(n+2)/2$

As so the number of dots on a double-250 set is $250\times 251\times 252/2=\boxed{7906500}$

The number of dominoes is equal to adding up the integers from 1 to 250.

$1 + 2 + 3 + 4 ... + 250$

a simple way to do this is using the formula:

$S = \frac{n(n+1)}{2}$ where $S$ is the total and $n$ is the number of terms.

with n= 250 we get:

$S = \frac{250*251}{2}$

$S = \frac{62750}{2}$

$S = 31375$

To find the number of dots we use the equation:

Total number of dots = number of dominoes * the amount of each number

Total number of dots = 31375 * 252 = $\boxed{7 906 500}$