Question

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).

Answer 1

(a)
`6!*12!` (b)
`6!*12!` (c)
`3*!6!*5!*7!`

Explanation:

GIVEN: There are
`18` chairs in a row (marked
`1,2,\dots, 18`) to seat
`6` chemistry majors,
`5` AMS majors and
`7` 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
`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).

SOLUTION:

(a)

as chemistry major first
`6` seats, and rest
`12` will sit on last
`12` seats.

total number of possible seating
`=^6P_6*^(12)P_(12)`

`=(6!)/((6-6)!)*(12!)/((12-12)!)`

`=6!*12!`

(b)

as chemistry major last
`6` seats, and rest
`12` will sit on first
`12` seats.

total number of possible seating
`=^(12)P_(12)*^6P_6`

`=(6!)/((6-6)!)*(12!)/((12-12)!)`

`=6!*12!`

(c)

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

then total number of possible seating
`=3!(^6P_6)(^5P_5)(^7P_7)`

`=3*!6!*5!*7!`