Introduction
A permutation is an arrangement of things with the order being important. abcd is different
to bcda. A combination is a selection of things e.g. abcd is the collection of the four letters with
no relevance given to the order. If a team of eleven players are selected for a football
team out of a selection of 20 persons with no importance given to the positions then the selection is
combination. However if each position (goalkeeper, left fullback etc. ) is chosen seperately
then the selection is a permutation.
Symbols..
n P(r) or ^{n } P (r) = Permutation of n items taken r at a time
n C(r) or ^{n } C (r) = Combination of n items taken r at a time

Permutations and combinations
Permutations: (Without Repitition)
Permutation illustrates the number of ways to arrange elements in a definite order.
The number of permutations of n items taken all at a time =
n! = 1.2.3 .....n (N Factorial)
The number of permutations of n elements of same kind taken r at a time.
nP(r) = n ! /(nr)!
Example The number of permutations of four l
etters a,b,c,d arranged three at a time is 4! /(43)! = 4.3.2.1 / 1 = 24
a,b,c / a,b,d /a,c,b /a,c,d / a,d,b/ a,d,c
b,a,c / b,a,d /b,c,a /b,c,d / b,d,a /b,d,c
c,a,b /c,a,d / c,b,a / c,b,d / c,d,a / c,d,b
d,a,b / d,a,c / d,b,a/ d,b,c / d,c,a /d,c,b
Permutation Table (without Repetition) 
n 
r 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
2 
2 
0 
0 
0 
0 
0 
0 
0 
0 
3 
3 
6 
6 
0 
0 
0 
0 
0 
0 
0 
4 
4 
12 
24 
24 
0 
0 
0 
0 
0 
0 
5 
5 
20 
60 
120 
120 
0 
0 
0 
0 
0 
6 
6 
30 
120 
360 
720 
720 
0 
0 
0 
0 
7 
7 
42 
210 
840 
2520 
5040 
5040 
0 
0 
0 
8 
8 
56 
336 
1680 
6720 
20160 
40320 
40320 
0 
0 
9 
9 
72 
504 
3024 
15120 
60480 
181440 
362880 
362880 
0 
10 
10 
90 
720 
5040 
30240 
151200 
604800 
1814400 
3628800 
3628800 
Permutations: (With Repetition)
A Permutation with repetition is the number of ways to arrange
elements in a definite order allowing elements to be repeated.
The number of permutations with repetition of n items taken r at a time =
nP(r) (with repetition) = n ^{r}
Example The number of permutations of three
letters a,b,c arranged two at a time is 3 ^{2} = 9
a,a / a,b / a,c / b,b /b,a / b,c/ c,c /c,a / c,b
Permutation Table (With repetition) 
n 
r 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
2 
4 
0 
0 
0 
0 
0 
0 
0 
0 
3 
3 
9 
27 
0 
0 
0 
0 
0 
0 
0 
4 
4 
16 
64 
256 
0 
0 
0 
0 
0 
0 
5 
5 
25 
125 
625 
3125 
0 
0 
0 
0 
0 
6 
6 
36 
216 
1296 
7776 
46656 
0 
0 
0 
0 
7 
7 
49 
343 
2401 
16807 
117649 
823543 
0 
0 
0 
8 
8 
64 
512 
4096 
32768 
262144 
2097152 
16777216 
0 
0 
9 
9 
81 
729 
6561 
59049 
531441 
4782969 
43046721 
387420489 
0 
10 
10 
100 
1000 
10000 
10^{5} 
10^{6} 
10^{7} 
10^{8} 
10^{9} 
10^{10} 
Combinations:
A combination is the number of ways to arrange elements with
no definite order. The number of combinations of n elements of same kind taken r at a time.
Equals n C(r) = n ! / r !(nr)!
Example. The number of combinations of four letters a,b,c,d arrange three at a time is 4! / 3!(43)! = 4.3.2.1 / 6.1 = 4
a,b,c / a,b,d /a,c,d /b,c,d
Combination Table (without Repetition) 
n 
r 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
2 
1 
0 
0 
0 
0 
0 
0 
0 
0 
3 
3 
3 
1 
0 
0 
0 
0 
0 
0 
0 
4 
4 
6 
4 
1 
0 
0 
0 
0 
0 
0 
5 
5 
10 
10 
5 
1 
0 
0 
0 
0 
0 
6 
6 
15 
20 
15 
6 
1 
0 
0 
0 
0 
7 
7 
21 
35 
35 
21 
7 
1 
0 
0 
0 
8 
8 
28 
56 
70 
56 
28 
8 
1 
0 
0 
9 
9 
36 
84 
126 
126 
84 
36 
9 
1 
0 
10 
10 
45 
120 
210 
252 
210 
120 
45 
10 
1 
Combinations: with repetition
Combinations with repetition are the number of ways to arrange elements
with no definite order but the elements can be repeated. The number of combinations of n elements of same kind taken r at a time.
Equals
n C(r) = (n + r 1)! / r ! (n  1)!
Example. The number of combinations of four letters a,b,c arrange
two at a time is (3 + 2  1)! / (2)!.(31)! = 4.3.2.1 / (2.1)(2.1) = 6
a,a /a,b / a,c / b,b / b,c/ c,c /
Combination Table (with repetition) 
n 
r 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
2 
3 
0 
0 
0 
0 
0 
0 
0 
0 
3 
3 
6 
10 
0 
0 
0 
0 
0 
0 
0 
4 
4 
10 
20 
35 
0 
0 
0 
0 
0 
0 
5 
5 
15 
35 
70 
126 
0 
0 
0 
0 
0 
6 
6 
21 
56 
126 
252 
462 
0 
0 
0 
0 
7 
7 
28 
84 
210 
462 
924 
1716 
0 
0 
0 
8 
8 
36 
120 
330 
792 
1716 
3432 
6435 
0 
0 
9 
9 
45 
165 
495 
1287 
3003 
6435 
12870 
24310 
0 
10 
10 
55 
220 
715 
2002 
5005 
11440 
24310 
48620 
92378 
