Important Formulas - Permutation and Combination - Tricks & Shortcuts

1.  Factorial Notation:

Let n be a positive integer. Then, factorial n, denoted n! is defined as:

n! = n(n - 1)(n - 2) ... 3.2.1.

Examples:

We define 0! = 1.

4! = (4 x 3 x 2 x 1) = 24.

5! = (5 x 4 x 3 x 2 x 1) = 120.

2. Permutations:

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

Examples:

i. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).

ii. All permutations made with the letters a, b, c taking all at a time are:

( abc, acb, bac, bca, cab, cba)

3. Number of Permutations:

Number of all permutations of n things, taken r at a time, is given by:

ⁿP_r = n(n - 1)(n - 2) ... (n - r + 1) =    n!/(n - r)!

Examples:

i. ⁶P₂ = (6 x 5) = 30.

ii. ⁷P₃ = (7 x 6 x 5) = 210.

iii. Cor. number of all permutations of n things, taken all at a time = n!.

4. An Important Result:

If there are n subjects of which p₁ are alike of one kind; p₂ are alike of another kind; p₃ are alike of third kind and so on and p_r are alike of r^th kind,

such that (p₁ + p₂ + ... p_r) = n.

Then, number of permutations of these n objects is =    n!/ ((p₁!).(p₂)!.....(p_r!))

5. Combinations:

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

Examples:i. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
Note: AB and BA represent the same selection.
ii. All the combinations formed by a, b, c taking ab, bc, ca.
iii. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
iv. Various groups of 2 out of four persons A, B, C, D are:1111111
AB, AC, AD, BC, BD, CD.
v. Note that ab ba are two different permutations but they represent the same combination.

6.  Number of Combinations:

The number of all combinations of n things, taken r at a time is:

ⁿC_r =       n!/( (r!)(n - r)!)  = (n(n - 1)(n - 2) ... to r factors)/ r!            .

               

Note:
i. ⁿC_n = 1 and ⁿC₀ = 1.

ii. ⁿC_r = ⁿC_{(n - r)}

Examples:

i.   ¹¹C₄ = (11 x 10 x 9 x 8)/ (4 x 3 x 2 x 1) = 330.

ii.   ¹⁶C₁₃ = ¹⁶C_{(16 - 13)} = ¹⁶C₃ = (16 x 15 x 14)/ 3! = (16 x 15 x 14)/ (3 x 2 x 1) =  560.

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