In how many ways a student can solve 12 questions of true /false with one answer

2 for the 1st
2 for the 2nd
2 for the 3rd
and so on, so by the product rule of counting the total is their product:

how many positive even integers less than 600 can be formed from 3, 4, 5, 6 ?

leading: 3
middle:4
least:2 
product: 24

a team of 4 is to be chosen at random from 5 women and 6 men.in how many ways can this be done if a. there are no restrictions. b. there must be more men than women.

We can use the counting principle:
a) 11 people, choose 4...11*10*9*8=7920
b) more men than women means 3 men... (6*5*4)(5*4)=2400

How many words of 4 letters can be formed with the letter a,b,c,d,e,f,g and h. If e & f are always include, letters are not repeated.

(2)(1)(6)(5)=60

A 6 character computer password is made up of 4 numbers followed by 2 letters. How many different passwords are possible?

counting principle applies: 
(10)(10)(10)(10)(26)(26)=6760000 

:)

How many combinations of license plates are there, if the license plate consists of 3 letters and 3 numbers where there is no specific order of which letter or number comes first. Repetition of letter/number is allowed. Example: AAA123 is one license plate A1A2A3 is another license plate

26)(26)(26)(10)(10)(10)=17576000

a tennis club consists of 10 boys and 8 girls. how many double teams(one boy and one girl) are possible?

choose boy: 10 
choose girl: 8 
choose both: 8*10=80 


:)

How many 3-digit odd numbers can be formed from the digits 1, 2, 4, 6, 8, 9 if repetition is allowed?

choose 1st/2nd digits: 6
third digit: 2
Total: (6)(6)(2)= 72 

:)

How many different committees of 3 people can be chosen to work on a special project from a group of 7 people?

this is a counting problem called a choose problem: 7 choose 3 (a combination). 

7 choose 3 = 

the number of ways of arranging 5 women and 4 men so that the row begin and end with women

Question 749359 in Permutations: the number of ways of arranging 5 women and 4 men so that the row begin and end with women 5-ways to choose 1st person 4-ways to choose the last (7)(6)(5)(4)(3)(2)(1) ways to choose middle people total: 20(7!)=10800 :)

Seven flute players are listed in the program in random order. How many different ways can the names be listed?

Assuming they all have different names 7! 
(counting principle) 


Take a look: 
7 ways to list first name
6 ways for the second
5 for the third
4 the fourth
3 fifth
2 sixth
1 last 
the total number of ways is the product: 
(7)(6)(5)(4)(3)(2)(1)=7! 

:)

There are seven finalists for two scholarships. Four of them are girls and three of them are boys. In how many ways can the scholarships be awarded to one boy and one girl?

This is a counting problem perfect for the counting principle. 
4 ways to choose a girl
3 ways to choose a boy 
ways to choose a girl and a boy: (4)(3)=12

In how many ways can three student-council members be elected from five candidates?

This is a counting problem so we apply the counting principle 
5 - choices for 1st electee
4 - remaining choices for second electee
3 - are left 
# of choices : (5)(4)(3)= 60 
60 ways 


:)

In how many ways can the word "TOPO" be arranged

The rule for repeated objects is 




(n!)/(m!*p!*...*q) 

so we get 
 

:)

You have invited your friend over for dinner. You have . How many meal choices can you offer?

Let's use the counting principle to make our computation. 

three kinds of fish, 
four types of vegetables,
two kids of rice, and 
3 pints (types?) of ice cream 


leads us to 


# of meal choices: (3)(4)(2)(3)=72 

:)