Chris flips twelve heads and four tails. That can be done in [imath]\dfrac{16!}{12!\cdot 4!}=1820[/imath] ways.
T T T T # ways to permute
0 0 4 4 4 10
0 1 3 4 4 60
0 2 2 4 4 30
0 ? ? ? ? ?
0 ? ? ? ? ?
1 ? ? ? ? ?
1 ? ? ? ? ?
1 ? ? ? ? ?
1 ? ? ? ? ?
2 ? ? ? ? ?
2 2 2 3 3 ?
TOT 320
I am wondering is there a faster solution. Since this is a Math Count question, the most time for it is maybe only 5 minutes.
q (-1)^q * choose(n,q) * choose(N - q*(r+1)+n-1, n-1)
0 1 * 1 * 1820 = 1820
1 -1 * 5 * 330 = -1650
2 1 * 10 * 15 = 150
SUM 320
T T T T # ways to permute
0 0 4 4 4 10
0 1 3 4 4 60
0 2 2 4 4 30
0 2 3 3 4 60
0 3 3 3 3 5
1 1 2 4 4 30
1 1 3 3 4 30
1 2 2 3 4 60
1 2 3 3 3 20
2 2 2 2 4 5
2 2 2 3 3 10
TOT 320