3. PERMUTATION
The different arrangement which can be made by taking
some or all of a number of thing are called permutation.
Suppose we have three objects a,b,c If we take two of
these at time, then the arrangement are
ab bc ca
ba cb ac
Thus the number of arrangement of 3 objects taken two at
a time is 6. Each of these is called a permutation.
4. NOTATION OF
PERMUTATION
• If n and r are positive integers such that 1
< r < n, then the number all permutation of
n distinct objects, taken r at a time is
denoted by the symbol nPr or P(n,r). Thus,
nPr or P(n,r)= Total number of permutation of
n distinct objects, taken r at a time.
5. Examples
• 6 permutation on a set of 3 letter taken 2
at a time i.e.,
3P2 = 6
• 24 permutation on a set of 4 objects taken 3
at a time i.e.
4p3 = 24
6. Factorial Notation
Recall the problem of counting how many ways we can seat three
men in three chairs
Because the product n x (n – 1) x … x 2 x 1 occurs often, we often
write it in shorthand notation as n!
The exclamation point is pronounced factorial
n! means the product of n down to 1
3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6
1! AND 0! are both equivalent to 1
n! = n · (n – 1)!
• We can expand a factorial into a product in order to quickly
evaluate expressions containing factorials
7. FUNDAMENTAL
PRINCIPLE OF COUNTING
• If you have 2 events: 1 event can occur m ways and another event
can occur n ways, then the number of ways that both can occur is
m*n
Event 1 = 4 types of meats
Event 2 = 3 types of bread
How many diff types of sandwiches can you make?
4*3 = 12
3 events can occur m, n, & p ways, then the number of ways all three
can occur is m*n*p
4 meats
3 cheeses
3 breads
How many different sandwiches can you make?
4*3*3 = 36 sandwiches
8. PERMUTATION OF ALIKE
THINGS
• The number of permutation of n things
taken all at a time where p of the things
are alike and of one kind, q there are alike
and of another kind, r others are alike and
of another kind and so on, is
n!
p! q! r!
10. COMBINATION
The different group or selection which can be made by taking
some or all of a number of things are called combination.
For examples
The combination which can be made by taking the letter a,b,c at a
time are
ab, bc, ac
Thus in combination we are only concerned with selection group
of things irrespective of the order of the things.
The number of all combination of n things taken r at a time is
denoted by nCr or c(n,r) or (n
r) clearly nCr is denoted only when n
and r are non-negative integers such that 0 < r < n.
11. EXAMPLE
• How many five-cards hands are possible from
a standard deck of cards?
Sol. 52C5 = 2598960
• Suppose there are 15 girls an 18 boys in a
class. In how many ways can 2 girls and 2
boys be selected for a group project?
Sol. 15C2 X 18C2 = 16065
