Let's think about *verbal arithmetic.*

The material of this puzzle is a summation of non-negative integers represented in a decimal format, for example, as follows.

905 + 125 = 1030

In verbal arithmetic, every digit appearing in the equation is masked with an alphabetic character. A problem which has the above equation as one of its answers, for example, is the following.

ACM + IBM = ICPC

Solving this puzzle means finding possible digit assignments to the alphabetic characters in the verbal equation.

The rules of this puzzle are the following.

- Each integer in the equation is expressed in terms of one or more digits '0'-'9', but all the digits are masked with some alphabetic characters 'A'-'Z'.
- The same alphabetic character appearing in multiple places of the equation masks the same digit. Moreover, all appearances of the same digit are masked with a single alphabetic character. That is, different alphabetic characters mean different digits.
- The most significant digit must not be '0', except when a zero is expressed as a single digit '0'. That is, expressing numbers as "00" or "0123" is not permitted.

There are 4 different digit assignments for the verbal equation above as shown in the following table.

Mask | A | B | C | I | M | P |
---|---|---|---|---|---|---|

Case 1 | 9 | 2 | 0 | 1 | 5 | 3 |

Case 2 | 9 | 3 | 0 | 1 | 5 | 4 |

Case 3 | 9 | 6 | 0 | 1 | 5 | 7 |

Case 4 | 9 | 7 | 0 | 1 | 5 | 8 |

Your job is to write a program which solves this puzzle.

The input consists of a number of datasets. The end of the input is indicated by a line containing a zero.

The number of datasets is no more than 100. Each dataset is formatted as follows.

N

STRING_{1}

STRING_{2}

...

STRING_{N}

The first line of a dataset contains an integer *N *
which is the number of integers appearing in the equation.
Each of the following *N * lines contains a string
composed of uppercase alphabetic characters 'A'-'Z'
which mean masked digits.

Each dataset expresses the following equation.

STRING_{1}+STRING_{2}+ ... +STRING_{N -1}=STRING_{N}

The integer *N * is greater than 2 and less than 13.
The length of *STRING* _{i} is greater than 0
and less than 9.
The number of different alphabetic characters appearing
in each dataset is greater than 0 and less than 11.

For each dataset, output a single line containing the number of different digit assignments that satisfy the equation.

The output must not contain any superfluous characters.

3 ACM IBM ICPC 3 GAME BEST GAMER 4 A B C AB 3 A B CD 3 ONE TWO THREE 3 TWO THREE FIVE 3 MOV POP DIV 9 A B C D E F G H IJ 0

4 1 8 30 0 0 0 40320

Source: ACM International Collegiate Programming Contest
, Japan Domestic Contest, Tokyo, Japan, 2009-07-03

http://www.waseda.jp/assoc-icpc2009/en/

http://www.waseda.jp/assoc-icpc2009/en/