Question

a) The 10 members of the committee have to choose one president and two vice presidents (these have to be three different people). In how many ways can they choose these officers? (Note: here is no distinction between the two vice presidents; there is not a “first” VP and a “second” VP).

b) Three couples go to a movie theater. They sit in consecutive seats such that each couple is seating together, that is each person is seating next to his/her partner. If there are 6 seats available, in how many ways can they sit?

c) A university wants to assign a three digit number to each classroom of a new building. They can use the digits {1,2,3,4,5} but they cannot use any digit more than once. How many classroom numbers can they assign if the numbers have to be less than 250?

d) The University wants to select 4 students for a feedback survey. They want all four students from either Prof. X’s section or Prof. Y’s section. Prof. X has 40 students in his class and Prof. Y has 30 students in his class. How many selections are possible?

Answer 1

a) Given, Total number of people=10

They have to select 3 people and there is no distinction between VP's, the number of selections possible are 10C3==>120.

b)Given, total number of seats are 6.

Number of people = 3 couples =>3*2=6 people

It is given that, the couples should be seated together. Hence, the number of seats favorable become 3(two seats are considered as 1).

Hence, there are 3 seats and 3 people(1 person =1 couple). Therefore , 3! ways are possible.

c)Given digits are {1,2,3,4,5}

Also, the numbers should be less than 250.

Hence, in the first position, the favorable digits are {1,2}

In the second position, the favorable digits are {1,2,3,4}

In the third position, the favorable digits are {1,2,3,4,5}

i.e.,, ___ ____ ___

2 4 5

Therefore, the number of classrooms they can assign if the numbers are less than 250 are 2*4*5 ==>40.

d)Given, number of students in prof.X section are 40

Number of students in prof.Y class are 30

To form committees of 4 people,we have to choose them from prof.X class or prof.Y class

Choosing from prof.X class, total people are 40 and we have to select 4 people ==>40C4

similarly, from prof.Y class 30C4

==> 40C4 or 30C4 ==> 40C4+30C4.