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Weekly Problem 30 - 2015

Stage: 3 and 4 Challenge Level: Challenge Level:1

Consider the following table:

 

The numbers in the first column are fixed and $a$, $b$, $c$ and $d$ should be chosen so that in the entire table the total number of $1$s is $a$, the total number of $2$s is $b$, the total number of $3$s is $c$ and the total number of $4$s is $d$.

Here is an example:








In the entire table there are two $1$s, three $2$s, two $3$s and one $4$.

How many other ways can you find to fill in the right-hand column? Can you find them and explain why there are no others?



This problem is taken from the UKMT Mathematical Challenges.

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