Factorial
The consecutive product of first n natural numbers is known as factorial n and is denoted as n! or Ðn
That is n! = 1 × 2 × 3 × 4 × 5 × ... × n
3! = 3 × 2 × 1
4! = 4 ×
3 ×
2 × 1
5! = 5 ×
4 ×
3 × 2 × 1
Also 5! = 5
× ( 4 × 3 × 2 × 1 )
= 5 × ( 4! )
Therefore this can be algebraically written as n! = n × (n – 1)! Note that 1! = 1 and 0! = 1.
Permutation:
Permutation means arrangement of things in different
ways. Out of three things
A, B, C taking
two at a time, we can arrange them in the following manner.
A B B A
A C C A
B C C B
Here we find 6 arrangements. In these
arrangements
order
of
arrangement
is
considered. The arrangement AB and the other arrangement BA are different.
The number of arrangements of the above is given as the number of permutations of 3 things taken 2 at a time which gives the value 6. This is written
symbolically, 3P2 = 6
Thus the number of arrangements that can be made out of n things taken r at a time is known as the number
of permutation of n things
taken r at a time and is denoted as
nPr.
The expansion of nPr is given below:
nPr =
n(n-1)(n-2) ……………[n – ( r – 1)]
The same can be written
in factorial notation
as follows:
nPr =
n! (n - r)!
For example, to find 10P3 we write this as follows:
10P3 = 10
(10 – 1) (10 – 2)
= 10 × 9 × 8
= 720
[To find 10P3, Start with 10, write the product of 3 consecutive natural numbers in the descending order]
Simplifying 10P3 using factorial
notation:
10 P3
10! 10´ 9 ´8 ´ 7 ´ 6 ´ 5 ´ 4 ´ 3 ´ 2 ´ 1
= =
(10 - 3)! 7
´ 6 ´
5 ´
4 ´
3 ´
2 ´
1
=
10 ´ 9´ 8
= 720
Note that nP0 = 1, nP1 = n, nPn = n!
Combinations:
A combination is a selection of objects without considering the order of arrangements.
For example, out of three things
A, B, C we have to select two things at a time. This can be selected
in three different ways as follows:
A B A C B C
Here the selection
of the object A B and B A are one and the same. Hence the order of arrangement is not considered in combination. Here the number of combinations from 3 different things taken 2 at a time is 3.
This is written symbolically 3C2 = 3
Thus the number of combination of n different things,
taken r at a time is given by
nCr = n Pr
r!
Or nCr =
n!
(n - r)! r!
Note that nC0 = 1, nC1 = n, nCn = 1
Find 10C3 , = 10 ´
9 ´8/ 3x2 = 120
3
Find 8C4 , 8C4
(8 ´ 7 ´ 6 ´ 5) / 1x2x3x4 = 70
Combinations:
A combination is a selection of objects without considering the order of arrangements.
For example, out of three things
A, B, C we have to select two things at a time. This can be selected
in three different ways as follows:
A B A C B C
Here the selection
of the object A B and B A are one and the same. Hence the order of arrangement is not considered in combination. Here the number of combinations from 3 different things taken 2 at a time is 3.
This is written symbolically 3C2 = 3
Thus the number of combination of n different things,
taken r at a time is given by
nCr = n Pr
r!
Or nCr =
n!
(n - r)! r!
Note that nC0 = 1, nC1 = n, nCn = 1
Find C , C
= 10 P3
10 ´
9 ´8
=
= 120
10 3
10 3
3! 1´ 2 ´
3
Find 8C4 , 8C4
8 ´ 7 ´ 6 ´ 5
= = 70
1 ´ 2 ´ 3 ´ 4
[ To find 8C4 : In the numerator, first write the product of 4 natural numbers starting
with 8 in descending order
and in the denominator write the factorial 4 and then simplify.]
Compare 10C8 and 10C2
10 C8
10 C2
10 ´ 9 ´ 8 ´ 7 ´ 6 ´ 5 ´ 4 ´ 3 10 ´ 9
= = = 45
1 ´ 2 ´ 3 ´ 4 ´ 5 ´ 6 ´ 7 ´ 8 1 ´ 2
10 ´ 9
= = 45
1 ´ 2
From the above,
we find 10C8 = 10C2
This can be got by the following method also:
10C8 = 10C(10 – 8) = 10C2
This method is very useful,
when the difference between n and r is
very high in nCr.
This property of the combination is written
as nCr = nC(n-r).
To find 200C198 we can use the above formula
as follows:
nCr = n! / (n - r)! r!
200 ´ 199/2 = 19900.
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