Question

In class of thirty two, twenty students are to be selected to represent the class. In how many ways can this be done, if :

(a) Paul refuses to represent the class?

(b) Michelle insists on representing the class?

(c) Jim and Michelle insist on representing the class?

(d) Either Jim or Michelle (or both) will represent the class?

(e) Paul and Michelle refuse to represent the class together?

How to solve this

Answer 1

Number of ways to select r items from n, nCr = n!/(r! X (n-r)!)

Number of students in class = 32

Number of students to be selected = 20

a) If Paul refuses to represent the class, 20 students are to be selected from 31 remaining students. Number of ways this can be done = 31C20

= 31!/(20! x 11!)

= 84,672,315

b) If Michelle insists on representing the class, remaining 19 students to represent the class must be selected from 31. Number of ways this can be done = 31C19

= 31/(19! x 12!)

= 141,120,525

c) If Jim and Michelle insist on representing the class, remaining 18 students can be selected from 30. Number of ways this can be done = 30C18

= 30!/(18! x 12!)

= 86,493,225

d) If either Jim or Michelle will represent the class, number of ways this can be done = Number of ways to select any 20 students from 32 - Number of ways 20 students can be selected such that both Jim and Michelle are not selected

= 32C20 - 30C20

= 225,792,840 - 30,045,015

= 195,747,825

e) if Paul and michelle refuse to represent, number of ways 20 students can be selected = 30C20

= 30,045,015