Question 924409
what i think is:


if x and y have to occupy the third and fourth slot, that leaves 7 slots for the other 10 people.


instead of 12P9, you get 10P7 * 2P2


10P7 = 10*9*8*7*6*5*4 = 603800


2P2 = 2*1 = 2


10P7 * 2P2 = 603800 * 2 = 1209600


the 10P7 takes care of the number of arrangements for the 10 remaining people in 7 remaining slots.


the 2P1 takes care of the number of arrangements for the 2 people in 2 slots.


to see this with smaller numbers, consider 6 people for 4 slots where x and y have to be in position 1 and 2.


instead of 6P4 = 360 possible arrangements, you have 2P2 * 4P2 = 24 possible arrangements.


let the people be xy1234


x is a person, y is a person, 1 is a person, 2 is a person, etc.


x and y have to be in positions 1 and 2, so those positions are filled.


the possible positions are:


xy12
xy13
xy14
xy23
xy24
xy34

yx12
yx13
yx14
yx23
yx24
yx34

xy21
xy31
xy41
xy32
xy42
xy43

yx21
yx31
yx41
yx32
yx42
yx43


that's a total of 24.


the formula appears to work so it can be applied to the larger problem.


the solution appears to be:


10P7 * 2P2 = 1209600






x and y have to be in positions 2 and 3
that leaves only 1 position for a and b and c



the possible arrangements are:


axy
bxy
cxy
ayx
byx
cyx


there are no other possible arrangements.