MouseHunt Math Quiz #1: Combinatorics

Question:
There are P possibilities that ten different hunters are hunting ten different mice.
There are Q possibilities that ten different hunters are hunting ten identical mice.
P+Q = ?

Example:
There are 20 possibilities that 3 different hunters are hunting 3 identical mice.
There are 64 possibilities that 3 different hunters are hunting 3 different mice (see below).

3 identical mice and 3 different hunters, namely (A, B, C).
possibility 1: (0, 0, 0) no mouse is caught at all
possibility 2: (1, 0, 0) hunter A catches a mouse
possibility 3: (0, 1, 0) hunter B catches a mouse
possibility 4: (0, 0, 1) hunter C catches a mouse
possibility 5: (1, 1, 0) hunter A and B catch a mouse each
possibility 6: (1, 0, 1) hunter A and C catch a mouse each
possibility 7: (0, 1, 1) hunter B and C catch a mouse each
possibility 8: (1, 2, 0) hunter A catches a mouse and hunter B catches 2 mice
possibility 9: (1, 0, 2) hunter A catches a mouse and hunter C catches 2 mice
possibility 10: (2, 1, 0) hunter B catches a mouse and hunter A catches 2 mice
possibility 11: (0, 1, 2) hunter B catches a mouse and hunter C catches 2 mice
possibility 12: (2, 0, 1) hunter C catches a mouse and hunter A catches 2 mice
possibility 13: (0, 2, 1) hunter C catches a mouse and hunter B catches 2 mice
possibility 14: (2, 0, 0) hunter A catches two mice
possibility 15: (0, 2, 0) hunter B catches two mice
possibility 16: (0, 0, 2) hunter C catches two mice
possibility 17: (3, 0, 0) hunter A catches all the mice
possibility 18: (0, 3, 0) hunter B catches all the mice
possibility 19: (0, 0, 3) hunter C catches all the mice
possibility 20: (1, 1, 1) hunter A, B and C catch a mouse each

3 different mice (x, y, z) and 3 different hunters (A, B, C).
possibility 1: no mouse is caught at all
possibility 2: hunter A catches mouse x
possibility 3: hunter A catches mouse y
possibility 4: hunter A catches mouse z
possibility 5: hunter B catches mouse x
possibility 6: hunter B catches mouse y
possibility 7: hunter B catches mouse z
possibility 8: hunter C catches mouse x
possibility 9: hunter C catches mouse y
possibility 10: hunter C catches mouse z
possibility 11: hunter A catches mouse x and y
possibility 12: hunter A catches mouse x and z
possibility 13: hunter A catches mouse y and z
possibility 14: hunter B catches mouse x and y
possibility 15: hunter B catches mouse x and z
possibility 16: hunter B catches mouse y and z
possibility 17: hunter C catches mouse x and y
possibility 18: hunter C catches mouse x and z
possibility 19: hunter C catches mouse y and z
possibility 20: hunter A catches mouse x and hunter B catches mouse y
possibility 21: hunter A catches mouse x and hunter B catches mouse z
possibility 22: hunter A catches mouse y and hunter B catches mouse x
possibility 23: hunter A catches mouse y and hunter B catches mouse z
possibility 24: hunter A catches mouse z and hunter B catches mouse x
possibility 25: hunter A catches mouse z and hunter B catches mouse y
possibility 26-31: hunter A and hunter C catches one mouse each
possibility 33-37: hunter B and hunter C catches one mouse each
possibility 38-40: a hunter catches all the mice
possibility 41-43: hunter A catches two mice and hunter B catches the last one
possibility 44-46: hunter A catches two mice and hunter C catches the last one
possibility 47-49: hunter B catches two mice and hunter A catches the last one
possibility 50-52: hunter B catches two mice and hunter C catches the last one
possibility 53-55: hunter C catches two mice and hunter A catches the last one
possibility 56-58: hunter C catches two mice and hunter B catches the last one
possibility 59-64: each hunter catches a mouse

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Solution:

This quiz is all about combinatorics. Read more here:
http://en.wikipedia.org/wiki/Combinatorics

P = 11^10
P is quite easy. Each mouse has 11 possibilities (not caught at all, caught by hunter#1, .., caught by hunter#10). 10 mice -> 11^10

Q = Combination (20, 10) = Factorial (20) / (Factorial (10)*Factorial(10)) = (20*19*18*17*..*11) / (10*9*8*...*1)

http://en.wikipedia.org/wiki/Stars_and_bars_(probability)

It is like you have 10 stars (10 mice) and 10 bars (10 hunters + one fake hunter for mice that are not caught at all).