1. 
P_j(n)={All permutations pi in P(n) such that pi[1]=j, j=1..n}
Example: 
P(3) = {123,132,213,231,312,321}:
P_1(3) = {123,132} in bijection with {23,32}
P_2(3) = {213,312} in bijection with {23,32}
P_3(3) = {321,231} in bijection with {23,32}

P_j(n) is in bijection with P(n-1)
|P_j(n)|=(|P_1(n)| + |P_2(n)| + ... + |P_n(n)|
p(n)=n*|P_1(n)| = n*|P(n-1)| = n!

2. 

P(1;1,2,3) = {1} p(1)=1
P(2;1,2,3) = {11,2} p(2) = 2
P(3;1,2,3) = {3,111,12,21} p(3) = 4
P(4;1,2,3) = {12,1111,112,121,22,211,3} p(4) = 7
|P(n;a1,a2,a3)| = p(n-1)+p(n-2)+p(n-3)


4. 
Someone is clever(C) and good looking(L)
Included twice in line 1
Excluded once in line 2
Included no times in line 3
Excluded no times in line 4
2-1+0-0=1

Some is clever(C), good looking(L), strong(S), and kind-hearted(K)
Included four times in line 1
Excluded six times in line 2
Included four times in line 3
Excluded once in line 4
4-6+4-1=1