Question

Solve the following using tree diagrams. Please draw the diagram!

A six person committee composed of Abba, Bull, Char,Dave, Eric and Ford is to select officer positions of chairman, secretary and treasurer. How many selections are there in which either Ford is an officer and Char is a secretary, or Abba is not chairman and Char is not an officer?

Answer 1

As there are 4 positions to be chosen from 6 people, total number of ways to do so without any restrictions is computed as:
= Permutation of 6 people taken 4 at a time
= 6*5*4*3

= 360 ways

Now total ways to have them 4 positions such that Ford is an officer and Char is secretary is computed here as:
= Number of ways to choose for other 2 positions from the leftover 4 people
= 4*3 = 12 ways

Total ways to have 4 positions such that Abba is not chairman and char is not an officer is computed here as:
= Total ways to select Chairman from 4 people ( excluding Abba and Char) * Total ways to select officer from 4 people (excluding char and 1 already selected ) * Total ways to choose rest 2 positions from remaining 4 people + Selecting Char as chairman * Total ways to select rest 3 positions from 5 people
= 4*4*4*3 + 5*4*3 = 96 + 60 = 156 ways

Total ways to select 4 positions such that   Ford is an officer and Char is secretary and Abba is not chairman and char is not an officer
= Number of ways to select Chairman from rest 2 * Number of ways to select Officer from remaining 2 + Number of ways to select officer from remaining 3

= 2*2 + 3 = 7 ways

Therefore total ways to select here using addition law of probabiltiy for two events is computed as:

= 12 + 156 - 7 = 161

Therefore there are 161 ways to make the selection here.

I hope it will help you!