Permutation and Combination Concepts and Formulas

Points to Remember:

1) Factorial:

The factorial can be defined as the product of the number (for which we have to find factorial) by its successor till it reaches to one.

We can write it as, n! (Factorial of n) = n (n-1) (n-2)....1

For example: The factorial of 3:

3! = 3*2*1 = 6

Note: The factorial of zero (0) is always 1 because an empty set can arrange in one way only.

2) Permutation: It refers to the number of ways a particular set can be arranged, where order of the arrangement matters. A combination lock can be called a permutation lock.

For example:

i) Let we have three letters a, b, and c and we have to arrange two letters at a time.

So, in this case, the permutations of the two letters = ab, ba, bc, cb, ac, and ca.

ii) If we have to arrange all letters (a,b,c) simultaneously, the permutation would be: abc, acb, bac, bca, cab, and cba.

Formula for calculating number of possible permutations of r things, from a set of n at a time is as follows:
ⁿP_r = n(n - 1)(n - 2)(n - 3)... (n - r + 1) =

For example:

i. ⁸P₃ = = = (8*7*6) = 336

ii. ⁷P₅ = = = 2520

iv. The number of permutations or arrangements of all n things at a time = n! (Factorial of n).
i.e., n = 3, so 3! = 3*2*1 = 6 (the number of permutations = 6)

3) Combinations:

It refers to the number of ways a particular set can be arranged, where order of the arrangement does not matter which means for a combination of the n number of things there may be different orders.

Formula for calculating the possible combination for r things, from a set of n objects at a time is as follows:

ⁿC_r = =

Note:

i) ⁿC_n = 1

ii) ⁿC₀ = 1

iii) ⁿC_r = ⁿC_(n-r)

For examples:

i. ⁸C₃ = = = = (8*7) = 56

Or, ⁸C₃ = ⁸C_(8-3) = ⁸C₅ = = 8*7 = 56

ii. ⁷C₅ = ⁷C_(7-5) = ⁷C₂ = = 21

Note:

• When the number of things is x, y, and z then the number of combinations taking two at a time will be xy, yz, and zx.

Note: xy and yx are same in the combination.

• The combination of all things at a time is xyz.