Page 189 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 13.5 Summary

 An arrangement of r objects, without considering order and without replication, chosen
from n different objects is known as a combination of n objects taken r at a time.

 The word selection is used when the order of things has no significance.
 A combination of n objects taken r at a time is a selection of r objects from a total of n
objects (r  n), where order does not matter.

 n  

 The notation for combination is usually   and read, “n choose r.”
r
  
! n
 The number of combinations is indicated by C  .
n r
r
(n r )! !

 The dissimilarity among combinations and permutations is in combinations you are
counting groups and in permutations you are counting different methods to assemble
items with respect to order.
 Number of combinations of ‘n’ dissimilar things taken ‘r’ at a time, when ‘p’ particular
things are forever included = C .
n–p
r–p
 Number of combination of ‘n’ dissimilar things, taken ‘r’ at a time, when ‘p’ particular
n-p
things are forever to be excluded = C
r
13.6 Keywords

Combination: An arrangement of r objects, without considering order and without
replication, chosen from n different objects is known as a combination of n objects taken r at a
time.

Selection: The word selection is used when the order of things has no significance.

13.7 Review Questions

1. In a class of 10 students, how many ways can a club of 4 students be arranged?
2. Find the number of ways to take 4 people and place them in groups of 3 at a time where
order does not matter.
3. A poker hand consists of 5 cards dealt without replacement and without regard to order
from a deck of 52 cards.
(a) Show that the number of poker hands is 2,598,960.
(b) Find the probability that a random poker hand is a full house (3 cards of one kind
and 2 of another kind).
(c) Find the probability that a random poker hand has 4 of a kind.
4. Suppose that in a group of n people, each person shakes hands with every other person.
Show that there are C(n, 2) different handshakes.
5. Find the number of poker hands that are void in at least one suit.
6. 3 cards are drawn from a standard deck of 52 cards. How many different 3-card hands can
possibly be drawn?

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