Tutorial 4

1. In how many ways can a committee of one president, one vice president, a secretary and a treasurer be chosen from five men and seven women?

2. A committee of four is chosen from five teachers and three mathematicians. In how many ways can this be done so that the committee contains

(i) at least one teacher?

(ii) at least one teacher and one mathematician?

3. Five children are to be seated on a bench. Find ( i ) how many ways the children can be seated , ( ii ) how many arrangements are possible if the youngest child is to sit at the left-hand end of the bench.

4. A playoff between two teams consists of at most five games . The first team that wins three games wins the playoff . In how many different ways can the playoff occur ?

5. An athlete owns six pairs of running shoes. How many different ways are of selecting

(i) 4 shoes?

(ii) 2 left shoes and 2 right shoes?

(iii) 2 pairs of shoes? 6. Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have the same number of men and women?

7. A concert pianist agrees to take part in a charity concert. She offers a choice of 10 works, of which 5 were composed by Chopin, 3 by Liszt and 2 by Schuamann. The concert committee is to select 4 of these works. Calculate the number of ways these 4 works can be selected if

(i) there is no restriction,

(ii) there are 2 works by Chopin, 1 by Liszt and 1 by Schumann,

(iii) there must be at least one work by each composer,

(iv) the selection contains works by exactly two of the three composers.

8. How many strings of six English letters are there ( a ) that contain no vowels , if letters can be repeated ? ( b ) that contain no vowels , if letters cannot be repeated ? ( c ) that start with a vowel , if letters can be repeated ? ( d ) that contain at