• Combinatorics And Graph Theory
• Combinatorics Formulas Pdf Converter

• In English we use the word 'combination' loosely, without thinking if the order of things is important. In other words: 'My fruit salad is a combination of apples, grapes and bananas' We don't care what order the fruits are in, they could also be 'bananas, grapes and apples' or 'grapes, apples.
• May 26, 2017  Permutations and combinations formulas pdf covers the important formulas for CAT. Download this pdf to score high in permutations and combinations in CAT. This Permutations and combinations formulas for CAT pdf will be very much helpful for CAT aspirants as significant number of questions are asked every year on this topic.
• COMBINATORICS nn! 01 11 22 36 424 5 120 6 720 7 5040 8 40320 9 362880. Expression 0! Is deflned to be 1 to make certain formulas come out simpler. The flrst few values of this function are shown in Table 3.3. The reader will note that this function grows very rapidly. The expression n.
• Combinatorics 3.1 Permutations Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and combinations. We consider permutations in this section and combinations in the next section.

Detailed tutorial on Basics of Combinatorics to improve your understanding of Math. Also try practice problems to test & improve your skill level. In these tutorials, we will cover a range of topics, some which include: independent events, dependent probability, combinatorics, hypothesis testing, descriptive statistics, random variables.

Combinatorics And Graph Theory

Combinatorics Formulas Pdf Converter

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:

   1. We define 0! = 1.

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

   3. 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:

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

   2. 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:

   nPr = n(n - 1)(n - 2) ... (n - r + 1) =	n!
   	(n - r)!

   Examples:

   1. ⁶P₂ = (6 x 5) = 30.

   2. ⁷P₃ = (7 x 6 x 5) = 210.

   3. 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)!.....(pr!)

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:

   1. 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.

   2. All the combinations formed by a, b, c taking ab, bc, ca.

   3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.

   4. Various groups of 2 out of four persons A, B, C, D are:

      AB, AC, AD, BC, BD, CD.

   5. Note that abba 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:

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

   Note:

   1. ⁿC_n = 1 and ⁿC₀ = 1.

   2. ⁿC_r = ⁿC_{(n - r)}

   Examples:

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

   ii. 16C13 = 16C(16 - 13) = 16C3 =	16 x 15 x 14	=	16 x 15 x 14	= 560.
   	3!		3 x 2 x 1