**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:

^{n}P_{r}= n(n - 1)(n - 2) ... (n - r + 1) = n!/(n - r)!**Examples:**

i.

^{6}P_{2}= (6 x 5) = 30.
ii.

^{7}P_{3}= (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*p*are alike of one kind;_{1}*p*are alike of another kind;_{2}*p*are alike of third kind and so on and_{3}*p*are alike of_{r}*r*kind,^{th}
such that (p

_{1}+ p_{2}+ ... p_{r}) = n.
Then, number
of permutations of these

*n*objects is = n!/ ((p_{1}!).(p_{2})!.....(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:**.**

^{n}C_{r}= n!/( (r!)(n - r)!) = (n(n - 1)(n - 2) ... to r factors)/ r!**Note:**

i.

^{n}C

_{n }= 1 and

^{n}C

_{0}= 1.

ii.

^{n}C_{r }=^{n}C_{(n - r)}**Examples:**

i.

^{11}C_{4}= (11 x 10 x 9 x 8)/ (4 x 3 x 2 x 1) = 330.
ii.

^{16}C_{13}=^{16}C_{(16 - 13)}=^{16}C_{3 }= (16 x 15 x 14)/ 3! = (16 x 15 x 14)/ (3 x 2 x 1) = 560.**Quantitative Aptitude Quizzes and Study Material**

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