I want everything on permutation and combination

Permutations and combinations are fundamental concepts in combinatorics, dealing with the arrangement and selection of objects from a set. Factorials Before diving into permutations and combinations, it's essential to understand factorials. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. n! = n × (n-1) × (n-2) × × 3 × 2 × 1 By definition, 0! = 1. Example: 5! = 5 × 4 × 3 × 2 × 1 = 120 Permutations Permutations refer to the arrangement of objects where the order matters. If you change the order of the objects, it's considered a different permutation. 1. Permutations of n distinct objects taken r at a time The number of permutations of n distinct objects taken r at a time is given by the formula: P(n, r) = (n!)/((n-r)!) Example 1: Arranging a subset How many ways can you arrange 3 different books on a shelf from a selection of 5 different books? Here, n=5 (total books) and r=3 (books to arrange). P(5, 3) = (5!)/((5-3)!) = (5!)/(2!) = (5 × 4 × 3 × 2 × 1)/(2 × 1) = 5 × 4 × 3 = 60 There are 60 ways to arrange the books. 2. Permutations of n distinct objects taken n at a time This is a special case where r=n. The formula simplifies to: P(n, n) = (n!)/((n-n)!) = (n!)/(0!) = (n!)/(1) = n! Example 2: Arranging all objects How many different ways can the letters of the word "MATH" be arranged? Here, n=4 (total letters) and r=4 (all letters are used). P(4, 4) = 4! = 4 × 3 × 2 × 1 = 24 There are 24 different arrangements. 3. Permutations with Repetition (Identical Objects) If there are n objects where n_1 are of one type, n_2 are of a second type, ..., n_k are of a k-th type, and n_1 + n_2 + + n_k = n, then the number of distinct permutations is: (n!)/(n_1! n_2! n_k!) Example 3: Arranging letters with repetition How many distinct arrangements can be made from the letters of the word "MISSISSIPPI"? Here, n=11 (total letters). M: 1 (n_1=1) I: 4 (n_2=4) S: 4 (n_3=4) P: 2 (n_4=2) Number of distinct arrangements = (11!)/(1! 4! 4! 2!) = (39,916,800)/(1 × 24 × 24 × 2) = (39,916,800)/(1152) = 34,650 There are 34,650 distinct arrangements. Combinations Combinations refer to the selection of objects where the order does not matter. If you select a group of objects, changing the order in which you picked them does not create a new combination. 1. Combinations of n distinct objects taken r at a time The number of combinations of n distinct objects taken r at a time is given by the formula: C(n, r) = nr = (n!)/(r!(n-r)!) Note that C(n, r) = (P(n, r))/(r!). This shows that for every r! permutations, there is only 1 combination. Example 4: Selecting a committee A committee of 3 people is to be chosen from a group of 7 people. How many different committees can be formed? Here, n=7 (total people) and r=3 (people to choose). The order of selection does not matter for a committee. C(7, 3) = (7!)/(3!(7-3)!) = (7!)/(3!4!) = (7 × 6 × 5 × 4 × 3 × 2 × 1)/((3 × 2 × 1)(4 × 3 × 2 × 1)) = (7 × 6 × 5)/(3 × 2 × 1) = 7 × 5 = 35 There are 35 different committees that can be formed. Example 5: Selecting items from a menu A student needs to choose 2 subjects from a list of 4 optional subjects. How many ways can the student choose the subjects? Here, n=4 (total subjects) and r=2 (subjects to choose). The order of choosing subjects does not matter. C(4, 2) = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4 × 3 × 2 × 1)/((2 × 1)(2 × 1)) = (12)/(2) = 6 There are 6 ways to choose the subjects. Key Difference: Order Matters vs. Order Does Not Matter Permutations: Use when the order or arrangement* of the selected items is important (e.g., arranging books, forming a password, assigning positions). Combinations: Use when only the selection or group* of items is important, and their order does not change the outcome (e.g., choosing a committee, selecting lottery numbers, picking cards).