Matematika Diskrit Soal ke-8


 

A computer programming team has 13 members.

a. How many ways can a group of seven be chosen to work on a project?

b. Suppose seven team members are women and six are men.

(i) How many groups of seven can be chosen that contain at four women and three men?

(ii) How many groups of seven can be chosen that contain at least one man?

(iii) How many groups of seven can be chosen that contain at most three women?

c. Suppose two team members refuse to work together on projects. How many groups of seven can be chosen to work

on a project?


Jawab :

1 team ada 13 anggota.

a. Banyak cara jika dibentuk sebuah group  yang dipilih hanya 7 orang untuk mengerjakan

project = 13C7 = 1716 cara.

b. Andaikata dalam 1 group beranggotakan 7 wanita dan 6 pria

(i). Banyak cara jika dalam group terdiri dari 4 wanita dan 3 pria = 7C4 * 6C3 =

  •  = 35 * 20 = 700 cara.

(ii) Banyak cara jika dalam group setidaknya ada satu pria :

1 Pria 6 Wanita = 6C1 * 7C6 = 6 * 7 =       42

2 Pria 5 Wanita = 6C2 * 7C5 = 15 * 21 = 315

3 Pria 4 Wanita = 6C3 * 7C4 = 20 * 35 = 700

4 Pria 3 Wanita = 6C4 * 7C3 = 15 * 35 = 525

5 Pria 2 Wanita = 6C5 * 7C2 = 6 * 21 =   126

6 Pria 1 Wanita = 6C6 * 7C1 = 1 * 7 =          7    +

Total =  1715 cara.

 

6C1 = 6 7C6 =(proses)  = 6
6C2 =(proses)= 15 7C5 = (proses) = 21
6C3 =(proses)  = 20 7C4 =(proses) = 35
6C4 =(proses)  = 15 7C3 =(proses)  = 35
6C5 =(proses)  = 6 7C2 =(proses)  = 21
6C6 = 1 7C1=  (proses)= 7

 

(iii) Banyak cara jika dalam group paling banyak 3 wanita :

3 Wanita 4 Pria = 7C3 * 6C4 = 35 * 15 = 525

2 Wanita 5 Pria = 7C2 * 6C5 = 21 * 6 =   126

1 Wanita 6 Pria = 7C1 * 6C6 = 7 * 1 =          7   +

Total   =      658 cara

7C3 =(proses)  = 35 6C4 =(proses)  = 15
7C2 = (proses) = 21 6C5 =(proses)  = 6
7C1 = 7 6C6 = 1

c. Andaikata 2 anggota menolak untuk bekerja, banyak cara untuk membentuk sebuah group

yang terdiri dari 7 orang = 11C7 =  = 330 cara.

Note : (proses) maksudnya adalah mencari hasil, maaf teman-teman karena format matematikanya tidak kebaca (^_^)

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