There are 18 chairs in a row (marked 1,2,..., 18) to seat 6 chemistry majors, 5 AMS majors and 7 music majors. (There are no double majors.) Find the number of possible seatings for each of the following:(a). The Chemistry majors must sit in the first 6 seats.(b). The Chemistry majors can not sit in the first 6 seats.(c). The students with the same majors must sit in a "block" (meaning each major sits together).

GIVEN: There are [tex]18[/tex] chairs in a row (marked [tex]1,2,\dots, 18[/tex]) to seat [tex]6[/tex] chemistry majors, [tex]5[/tex] AMS majors and [tex]7[/tex] music majors. (There are no double majors).

TO FIND: Find the number of possible seating for each of the following:(a). The Chemistry majors must sit in the first [tex]6[/tex] seats.(b). The Chemistry majors can not sit in the first [tex]6[/tex] seats.(c). The students with the same majors must sit in a "block" (meaning each major sits together).

SOLUTION:

(a)

as chemistry major first [tex]6[/tex] seats, and rest [tex]12[/tex] will sit on last [tex]12[/tex] seats.

total number of possible seating [tex]=^6P_6\times^{12}P_{12}[/tex]

[tex]=\frac{6!}{(6-6)!}\times\frac{12!}{(12-12)!}[/tex]

[tex]=6!\times12![/tex]

(b)

as chemistry major last [tex]6[/tex] seats, and rest [tex]12[/tex] will sit on first [tex]12[/tex] seats.

total number of possible seating [tex]=^{12}P_{12}\times^6P_6[/tex]

[tex]=\frac{6!}{(6-6)!}\times\frac{12!}{(12-12)!}[/tex]

[tex]=6!\times12![/tex]

(c)

if the students with the same majors must sit in a block.

then total number of possible seating [tex]=3!(^6P_6)(^5P_5)(^7P_7)[/tex]

[tex]=3\times!6!\times5!\times7![/tex]