Chapter 2: Counting Problems
Department of Mathematics & Statistics
Wake Forest University
Chapter 2
Counting
Problem
How many ways can a committee be formed from 4 men and 6
women with:
(i) At least 2 men and atleast twice as many women as men?
(ii) Four members, at least 2 of which are women, and Mr. and
Mrs. Schmidt will not serve together?
Chapter 2
Counting
Problem
How many ways can a committee be formed from 4 men and 6
women with:
(i) At least 2 men and atleast twice as many women as men?
(ii) Four members, at least 2 of which are women, and Mr. and
Mrs. Schmidt will not serve together?
(i)
The committees can be of 4 types: (m, w ) = (2, 4) or (2, 5) or
(2, 6) or (3, 6). So the number of committees is
(C24 × C46 ) + (C24 × C56 ) + (C24 × C66 ) + (C34 × C66 ).
Chapter 2
Counting
Problem
How many ways can a committee be formed from 4 men and 6
women with:
(i) At least 2 men and atleast twice as many women as men?
(ii) Four members, at least 2 of which are women, and Mr. and
Mrs. Schmidt will not serve together?
(ii)
Let n1 be the number of 4 member committees, atleast 2 of which
are women and let n2 be the number of 4-member committees,
atleast 2 of which are women with Mr. and Mrs. Schmidt both
included. The number of committees is n1 − n2 .
n1 = (C24 × C26 ) + (C14 × C36 ) + (C04 × C46 )
n2 = (C13 × C15 ) + (C25 )
Chapter 2
Counting
Problem
In how many ways can 8 people be seated in a row if (a) there are
no restrictions on the seating arrangement?
(b) persons A and B must sit next to each other?
(c) there are 4 men and 4 women and no 2 men or 2 women can
sit next to each other?
(d) there are 5 men and they must sit next to each other?
(e) there are 4 married couples and each couple must sit together?
Chapter 2
Counting
(a)8! = 40, 320
(b)2 × 7! = 10, 080
(c)4!4!2
(d) 5!4! = 2, 880
(e)4!24 = 384
Chapter 2
Counting
In how many ways can 3 novels, 2 mathematics books, and 1
chemistry book be arranged on a bookshelf if
(a) the mathematics books must be together and the novels must
be together?
(b) the novels must be together, but the other books can be
arranged in any order? [2+2 points]
Chapter 2
Counting
Solution (a) 3!2!3! (b) 3!4!
Chapter 2