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Chapter 5

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16 formulas

Factorial of a Non-Negative Integer

Topic: Logical ReasoningChapter: Chapter 5
n!=n×(n−1)×(n−2)×⋯×2×1n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1
The factorial of a non-negative integer 'n', denoted by n!, is the product of all positive integers less than or equal to n. It represents the number of ways to arrange 'n' distinct items in a sequence. By definition, 0! = 1.

Permutation (Arrangement of 'r' items from 'n' distinct items)

Topic: Logical ReasoningChapter: Chapter 5
P(n,r)=nPr=n!(n−r)!P(n, r) = nP_r = \frac{n!}{(n-r)!}
This formula calculates the number of distinct arrangements (permutations) of 'r' items chosen from a set of 'n' distinct items, where the order of selection matters. It is used when the arrangement or sequence of items is important.

Combination (Selection of 'r' items from 'n' distinct items)

Topic: Logical ReasoningChapter: Chapter 5
C(n,r)=nCr=(nr)=n!r!(n−r)!C(n, r) = nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}
This formula calculates the number of distinct selections (combinations) of 'r' items chosen from a set of 'n' distinct items, where the order of selection does not matter. It is used when only the group or subset of items is important, not their arrangement.

Permutation with Repetition (Arrangement of 'r' items from 'n' distinct items with repetition allowed)

Topic: Logical ReasoningChapter: Chapter 5
nrn^r
This formula calculates the number of ways to arrange 'r' items chosen from 'n' distinct items, where each item can be chosen multiple times (repetition is allowed). For example, forming a password of length 'r' using 'n' distinct characters.

Permutation of 'n' objects where 'p' are of one kind, 'q' of another, etc.

Topic: Logical ReasoningChapter: Chapter 5
n!p!q!r!…\frac{n!}{p!q!r!\dots}
This formula calculates the number of distinct permutations of 'n' objects where some objects are identical. Here, 'n' is the total number of objects, and p, q, r, ... are the counts of identical objects of different kinds.

Circular Permutation of 'n' distinct objects

Topic: Logical ReasoningChapter: Chapter 5
(n−1)!(n-1)!
This formula calculates the number of ways to arrange 'n' distinct objects in a circle. Since there is no fixed starting or ending point in a circle, one position is fixed to avoid overcounting rotations as distinct arrangements.

Circular Permutation of 'n' distinct objects (when clockwise and anti-clockwise arrangements are considered same)

Topic: Logical ReasoningChapter: Chapter 5
(n−1)!2\frac{(n-1)!}{2}
This formula is used for circular permutations where flipping the arrangement results in an identical configuration (e.g., arranging beads on a necklace). In such cases, clockwise and anti-clockwise arrangements are not considered distinct.

Factorial

Topic: Time Value of MoneyChapter: Chapter 5
n!=n×(n−1)×(n−2)×...×2×1n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1
The factorial of a non-negative integer 'n', denoted by n!, is the product of all positive integers less than or equal to n. It represents the number of ways to arrange 'n' distinct items in a sequence. By definition, 0! = 1.

Permutation (without repetition)

Topic: Time Value of MoneyChapter: Chapter 5
P(n,r)=nPr=n!(n−r)!P(n, r) = {}_{n}P_{r} = \frac{n!}{(n-r)!}
This formula calculates the number of distinct arrangements (permutations) of 'r' items chosen from a set of 'n' distinct items, where the order of selection matters. 'n' is the total number of items available, and 'r' is the number of items to be arranged.

Combination (without repetition)

Topic: Time Value of MoneyChapter: Chapter 5
C(n,r)=nCr=n!r!(n−r)!C(n, r) = {}_{n}C_{r} = \frac{n!}{r!(n-r)!}
This formula calculates the number of distinct selections (combinations) of 'r' items chosen from a set of 'n' distinct items, where the order of selection does not matter. 'n' is the total number of items available, and 'r' is the number of items to be selected.

Permutation with Repetition (n^r)

Topic: Time Value of MoneyChapter: Chapter 5
nrn^r
This formula calculates the number of ways to arrange 'r' items from a set of 'n' distinct items, where repetition of items is allowed. For example, if you have 'n' choices for each of 'r' positions, the total number of arrangements is n multiplied by itself 'r' times.

Factorial

Topic: Fundamental Principles and FormulasChapter: Chapter 5
n!=n×(n−1)×(n−2)×⋯×2×1n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1
Represents the product of all positive integers less than or equal to n. It is used in permutations and combinations to count arrangements. By definition, 0! = 1.

Permutation of n distinct items taken r at a time

Topic: Fundamental Principles and FormulasChapter: Chapter 5
P(n,r)=nPr=n!(n−r)!P(n, r) = {}_{n}P_{r} = \frac{n!}{(n-r)!}
Calculates the number of distinct arrangements of 'r' items chosen from a set of 'n' distinct items, where the order of selection matters. Here, n \ge r \ge 0.

Combination of n distinct items taken r at a time

Topic: Fundamental Principles and FormulasChapter: Chapter 5
C(n,r)=nCr=n!r!(n−r)!C(n, r) = {}_{n}C_{r} = \frac{n!}{r!(n-r)!}
Calculates the number of distinct selections of 'r' items chosen from a set of 'n' distinct items, where the order of selection does not matter. Here, n \ge r \ge 0.

Permutation with Repetition

Topic: Fundamental Principles and FormulasChapter: Chapter 5
nrn^r
Calculates the number of arrangements when 'r' items are chosen from 'n' distinct items, and repetition is allowed. Each choice is independent.

Permutation of n items with identical items

Topic: Fundamental Principles and FormulasChapter: Chapter 5
n!p!q!r!\frac{n!}{p!q!r!}
Calculates the number of distinct arrangements of 'n' items where 'p' items are of one type, 'q' items are of another type, 'r' items are of a third type, and so on.
Chapter 5 Formulas — CA Foundation | Kartavvya