11. In the following expression each $\square$ is to be replaced with either + or - in such a way that the result of the calculation is 100 .
123
45
67
89

The number of + signs used is $p$ and the number of - signs used is $m$. What is the value of $p-m$ ?
A -3
B $\quad-1$
C 0
D 1
E 3
Solution: B
The sum is made up of 123 and $\pm 45, \pm 67$ and $\pm 89$. Suppose that the total of the positive terms in the calculation is $x$ and the total of the negative terms is $y$. So $x>0$ and $y<0$. We need to have that
$$
x+y=100 \text {. }
$$

We also have that
$$
x-y=123+45+67+89=324 \text {. }
$$

Adding these equations, we obtain $2 x=424$. So $x=212$ and hence $y=-112$. It is readily seen that $45+67=112$ and that no other combination of 45,67 and 89 gives a total of 112 . So the correct calculation must be $123-45-67+89=100$ with 1 plus sign and 2 minus signs. So $p=1$ and $m=2$, giving $p-m=-1$.