By placing addition signs between some of the digits of the number 123456789, we can obtain various sums. For example, we can obtain 279 with four addition signs: $123+4+56+7+89=279$. How many addition signs are needed to obtain the number 54 in this way?
A) 4
B) 5
C) 6
D) 7
E) 8
Solution

## ALTERNATIVE D

Since we want to obtain the sum of 54, we must place addition signs between all the digits starting from 5, that is, $1 ? 2 ? 3 ? 4 ? 5 + \underbrace{6+7+8+9}_{30} = 54$. Therefore, we need $1 ? 2 ? 3 ? 4 ? 5 = 24$.

Using the same argument as before, we see that this can only be done as $12 + 3 + 4 + 5$. Thus, $12 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 54$ is the desired expression, for which we need 7 addition signs.