COMBINATIONS
Combinations. We can choose k objects out of the collection of N objects in
k
N
ways (reads “N
choose k”). The order of the chosen objects is not important. We can reorder these k objects in k!
ways, therefore for each choice of k objects we can put them into a line in k! ways. Hence,
k
Nk
k
N
!
. Therefore,
1)1(
)1()1(
kk
kNNN
k
N
.
Exercises
1. Compute a) 102 b)
2
10
2. Compute a) 93 b)
3
9
c)
3
9
9
3
3. There are 9 objects to choose from. In how many ways one can а) make an ordered list of 3
objects? b) choose an unordered set of 3 objects (order is not important)?
4. There are 10 members of a team. In how many ways we can choose
a) a leader and his assistant? b) two persons on duty?
5. There are 10 members of a team. In how many ways we can choose 8 players for the next game?
6. Compute
6
9
. Explain why the answer coincides with one of the answers above.
7. In a card of the lottery “4 from 12” we should fill four numbered boxes out of twelve. During the
drawing one draws randomly four numbers out of twelve. The main prize wins the one who guesses
all numbers. How many cards one has to fill to win for sure the main prize?
Problems
8. Compute
2
1001
3
1001
9. In a card one has to mark four horses out of twelve horses participating in a race. A person wins
the prize “Million” if he marks those horses that come first. Peter knows two horses that for sure
will not come among the first four. How many cards should he fill to win the prize “Million” for
sure?
10. There are 15 members of a team including Peter. One has to choose 11 members for the next
game. In how many ways one can do it a) including Peter b) not including Peter?
