$A$ rolls a standard $6$ sided die $20$ times while $B$ rolls it $21$ times. Find the probability that $A$'s outcome is more than $B$'s. Here, outcome means the sum of the numbers appearing on all the rolls.
I tried to approach this in many ways, using mostly generating functions but the cases were just too many to handle.
I even get a hint of the multinomial distribution of the $20$ and $21$ rolls, but I didn't go far with that too.
Please help me out. Hints and answers appreciated. Thank you.