INDEPENDENCE has 1 C, 2 D's, 4 E's, 1 I, 3 N's, and 1 P
There are 6 cases:
Case 1: The 5 letter arrangements like the distinguishable
arrangements of VWWWW of which there are 5!/4! = 5.
Choose the W as E and the V any of the other 5 ways.
That's 1*5 = 5 to multiply by 5. So there are 25 arrangements for case 1.
Case 2: The 5 letter arrangements like the distinguishable
arrangements of VVWWW of which there are 5!/(2!3!) = 10.
We can only use D,E, and N in this case
Choose the W 2 ways,as E or N, and the V either of the remaining 2 ways.
That's 2*2 = 4 to multiply by 10. So there are 40 arrangements for case 2.
Case 3: The 5 letter arrangements like the distinguishable
arrangements of VWXXX of which there are 5!/3! = 20.
We can choose the X 2 ways, as E or N, then the V and W as any combination
of 2 letters from the remaining 5 letters, 5C2=10.
That's 2*10 = 20 to multiply by 20. So there are 400 arrangements for case 3.
Case 4: The 5 letter arrangements like the distinguishable
arrangements of VWWXX of which there are 5!/(2!2!) = 30.
We can choose the W and X as any combination of 2 from the 3 letters D,E,N,
3C2=3. Then we can choose the V as any one of the remaining 4 letters.
That's 3*4 = 12 to multiply by 30. So there are 360 arrangements for case 4.
Case 5: The 5 letter arrangements like the distinguishable
arrangements of VWXYY of which there are 5!/2! = 60.
We can choose the Y 3 ways, as D,E, or N. Then we can choose V, W and X
as any combination of 3 from the remaining 5 letters, 5C3=10
That's 3*10 = 30 to multiply by 60. So there are 1800 arrangements for case 5.
Case 6: The 5 letter arrangements like the distinguishable
arrangements of VWXYZ of which there are 5! = 120
We can choose those as any combination of 5 from the 6 letters C,D,E,I,N,P.
That's 6C5=6 to multiply by the 120.
That's 6 to multiply by the 120. So there are 720 arrangements for case 6.
Grand total for the 6 cases: 25+40+400+360+1800+720 = 3345
Edwin
