If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ yellow, but I can only pick three from each jar without replacement. How many events occur where there is the colour of the ball in the same position across the three jar selections, excluding events where there are multiple balls that are in the same position and colour across all three selections?
Situation as shown in this rough sketch
For Example: Jar 1:(Red, Blue, Red) Jar 2:(Yellow, Blue, Red) Jar 3:(Blue, Blue, Blue), this would be an even where a blue ball has been selected in the same position across all three jars.
Another Example: Jar 1:(Red, Blue, Red) Jar 2:(Yellow, Blue, Red) Jar 3:(Blue, Blue, Red), this would be an even where a blue ball and a red ball has been selected in the same position across all three jars and is excluded from the answer to the problem.