To find the total number of ways to distribute the 10 ice cream cones, we find 10! = 10*9*8*7*6*5*4*3*2*1 = 3628800. This is because the first child has 10 choices, the next has 9 choices since one cone was already taken, etc.
However, there are fewer than 3628800 distributions because some of the cones are identical. We need to divide the total number of ways to distribute 10 cones by the fractorial of all the counts of types of cones.
The counts are 2, 3, 4, and 1, so the number of distinct distributions is:
10!/(2!3!4!1!) = 12600
The answer is 12600.
