1 pt In a class of 100 students what is the smallest number

(1 pt) In a class of 100 students what is the smallest number of students such that their last names start with the same letter? (1 pt) What should be the minimum number of US students in a college to make sure that there are at least 100 students from the same US state? (1 pt) If the password locking TV consists of 4 digits, how many different passwords my son will have to try in the worst case to unlock the TV? (1 pt) There are 4 different candidates for a governor of a state. In how many different orders can the names of the candidates be printed on a ballot? (1 pt) How many different signals, each consisting of 6 flags hung in a vertical line, can be formed from 4 identical red flags and 2 identical blue flags? (1 pt) Find the number of ways that an organization consisting of 10 members can elect a president, a treasurer, and a secretary (one person may be elected just for one position). (1 pt) Find the number of ways that an organization consisting of 10 members can elect 3 representatives to a senate. (2 pts) How many terms does (x+y)9 have? What is the coefficient of the term x^3 y^6? (1 pt) Suppose that an ice-cream café has 10 different flavors of ice cream. In how many different ways one can choose 3 flavors of ice-cream, so that order of flavors does not matter? (There is an unlimited amount of each flavor of ice-cream.)

Solution

1.100=26*3+22 HENCE THE SMALLEST NO. OF STUDENTS IS 3

2.NO. OF US STATES=50

given that there are at least 100 students from the same US state

hence min. no. of students are 100*50=5000

3. given the password has 4 digits

each digit has ten chances i.e, 1,2,3,4,5,6,7,8,9,0

there fore total no. of chances is 10⁴

4. the total no.of names =4

the no. of possibilities for arranging them is 4!=24

6. the total no.of possibilities is 10P3=10!/(10-3)!=10!/7!=8*9*10=720