Question

(1) [20 pts] A club is considering changing its bylaws. In an initial vote on the issue, 12 of members of the club favored the change and 8 did not. A committee of 6 is to be chose from the 20 club members to devote further study on the issue. (a) [6 pts] How many committees of 6 can be formed? EXPLAIN! (b) [6 pts] How many committees of 6 can be formed if Donald and Hillary can not se on the same committee? EXPLAIN! (c) [8 pts) How many of the committees will contain at least 3 members who, in the preliminary survey, favored the change in the bylaws? EXPLAIN! utations.

Answer 1

1 (a) he number oF committees of 6 that can be Formed from the 20 club members is given as 20 20 (20-6)! 6! 14! 6 20X 19X 18 X 17X 16 X15 20 I 6 38760 Consider the number of commi ttees that can be formed with Donald and Hillary together in a committee. So out oF 6 members in a committee 2 are Fixed i.e Donald and Hillary, now choose the remaining 4 members For a committee From the remaining 18 members (excl- uding Donald and Hillary), this can be done in (b) 181 18 I C4 Hays .e. (18-4)I 4!141 X 4 18 X17X16X1!5 3060 Thus, the number of committees in which either no Donald and no Hillary or only Donald or only Hillary is present are given as

38 760 306035700 Also, it should be noted if we are looting for committees which have either Donald or Hillary but not both toge her in a committee, this can be done by, first selecting Donald and excluding Hillary,i.e. 5 mem have to be selected from 19-1 18 members, subtract 1 ezclude Hillary, So, the number of committees tho.t have Donald only and not Hilary are 18 Litewise, number oF committees that have Hillary only and not Donald ar again 's 5 So, the total number oF committees that can be formed with either Donald or Hillary in committee but not both togeher are 181 18 (18-5)lx5 5X4X3X2XJ 17136 (C) The committees that contain exactly 3 members who favored the change ih the bylaws is formed by selecting 3 members out of 12 and remaining 3 members aut of 8 i.e.

3 3 Likewise, committees that contain 4 members who Favorea change in bylaws are and Committes that contain 5 members who favored Change in bylaws are 12c v 8 5 and lastly committees that contain all 6 members who avored change in bylaws are Now, add them all to determine the number oF committees that contain atleast 3 members who favored the change in bylaws ar e 12 12 12 6 12X11X10x9 4X3 X2X1 3X2X 1 3X2 X1 12 X11 X10x 9 X9* 5x 4X3X 2X1 2x1 12X11X 10x9x8x7 33440O

hus, the number of commi thees that conrtain at least 3 members who favored the change in bylaws are 33440