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

Answer:

(a) $6!\times12!$ (b) $6!\times12!$ (c) $3\times!6!\times5!\times7!$

Step-by-step 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 $=^6P_6\times^{12}P_{12}$

                                                      $=\frac{6!}{(6-6)!}\times\frac{12!}{(12-12)!}$

                                                      $=6!\times12!$

(b)

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

total number of possible seating $=^{12}P_{12}\times^6P_6$

                                                      $=\frac{6!}{(6-6)!}\times\frac{12!}{(12-12)!}$

                                                      $=6!\times12!$

(c)

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

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

                                                              $=3\times!6!\times5!\times7!$