Combinatorics

1003024607

Level: 
A
On the shelf, there should be three blue cups, three red cups, two yellow cups and two green cups arranged in a row from the left to the right. The cups of the same color are not mutually distinguishable. How many arrangements of these cups are possible?
10!(2!)2(3!)2=25200
10!46!=1260
10!22!3!=151200
10!42!3!=75600

1003024604

Level: 
B
Without using a calculator, identify which of the following values is the greatest one.
the number of ordered arrangements of 5 objects taking 3 at a time
the number of unordered selections of 3 objects out of 5 different kinds with repetition allowed
the number of ordered arrangements of 5 objects in which 3 are identical
the number of unordered selections of 3 objects out of 5 objects

1003024601

Level: 
A
Assume the password for the safe deposit box consists of four different letters from the set {A;B;C;D;E;F;G;H} and four different numbers from the set {1;2;3;4;5;6;7}. How many different passwords are there?
(84)(74)8!=98784000
8!4!7!3!8!=56899584000
(8!4!+7!3!)8!=101606400
((84)+(74))8!=4233600