Multiplication of Any Two-Digit Number with Any Two-Digit Number, Such that Their Ten's Digits are Equal, and Their One's Digits Add Up to Ten

In this Math Trick we are going to multiply any two-digit number with any two digit number, Such that their ten's digits are equal, and their one's digits add up to ten. This is represented by the formula, (10A+B) (10A 4C) = ; A = (0,1...9), B+C = 10 Just multiply the ten's digit of the mulitiplicand plus one with the ten's digit of the second number and write it down as the first and/or second digits or thousand's and/or hundred's digits of the answer, and multiply the one's digits, and write it down as the third…

The Square of Any Two-digit Number

, In this Math Trick, we are going to square any two-digit number.   It is represented by, the formula, (10X + Y) 2  = 100X2 + 20XY + Y 2 Or by Square of Binomial Method, with HTO, HundredsTensOnes (1st2nd3rd-Digits) Let's solve for, 1. 12 2 , 2. 23 2  3. 87 2 Solution: 1a. 12 2  = 100*1 2  + 20*1*2 + 2 2              = 100 + 40 + 4               = 144 1b. By Square of Binomial Method : 12 2   = 1 2  for the first digit l 1*2*2 for the second digit l 2 2         = 1 l 4 l 4         = 144 2a. 23 2  = 100*2 2  + 20*2*3 + 3 2             =…

Multiplication of Two Same-Digits Number and One-Digit Number

1. 22 x 9 = 2. 33 x 8 = 3. 44 x 7 = 4. 55 x 6 = 1. 22 x 9 = ; 2 x 9 = 18. Write 1 & 8 with one-digit apart. They will be the first and third digits of the answer. Now, for the second digit, add the first and third digits, which in this example are 1 & 8. Their sum is 9. Therefore, the answer is 198. 2. 33 x 8 = ; 3 x 8 = 24. Write 2 & 4 with one-digit apart. They will be the first and third digits of the answer. Now, for the second digit, add the first and third digits, which in this example are 2 & 4. Their sum is 6. Therefo…

Whole Numbers: A Deep Dive into the Basics

Whole numbers are the foundation of mathematics, representing the basic counting numbers starting from zero and extending infinitely. They are non-negative integers, meaning they don't include fractions, decimals, or negative values. Let's explore their characteristics and significance:   Characteristics of Whole Numbers   - Non-negative: Whole numbers are always greater than or equal to zero (0, 1, 2, 3...). They do not include negative numbers (-1, -2, -3...). - Integers: Whole numbers are a subset of integers. Integers include …

Understanding Simple Probability: A Fun Guide!

The probability of a simple event is a fundamental concept in probability theory. It represents the likelihood of a specific outcome occurring in a single trial of an experiment where all outcomes are equally likely. Let's break down the key aspects:   1. Simple Events : A simple event is an outcome of an experiment that cannot be further broken down into smaller events. For example, if you roll a six-sided die, each individual outcome (rolling a 1, 2, 3, 4, 5, or 6) is a simple event. Flipping a coin and getting heads is another exa…

Mastering Combinations: Fun Quiz & Solutions!

Solve the following questions. These questions focus on different aspects of combinations, including calculating combinations, understanding their applications, and recognizing related concepts. I'll assume you're familiar with the basic formula for combinations: nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen.   1. Question: How many ways can you choose a committee of 3 people from a group of 10?   Answer: 120   Solution: This is a combination problem because the order in which the …

Mastering Combinations: Fun Quiz & Solutions!

Solve the following questions. These questions focus on different aspects of combinations, including calculating combinations, understanding their applications, and recognizing related concepts. I'll assume you're familiar with the basic formula for combinations: nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen.   1. Question: How many ways can you choose a committee of 3 people from a group of 10?   Answer: 120   Solution: This is a combination problem because the order in which the …

Mastering Factorials: Fun Quiz & Solutions!

A. Solve the following. Remember that the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. 0! is defined as 1.   Question 1: What is 6!?   Answer: 720   Solution:6! = 6 x 5 × 4 × 3 × 2 × 1 = 720   Question 2: Calculate 9!.   Answer: 362,880   Solution: 9! = 9 x 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880   Question 3: Simplify 10!/7!.   Answer: 90   Solution: 10!/7! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1/(7 × 6 × 5 × 4 × 3 × 2 × 1) = 10 × 9 x 8 = 720   Question 4: Evaluate 7…

Why 0! = 1 and 1! = 0!

First, let's define factorial (!) or factorial notation (!). Factorial notation (!) is shorthand for multiplying consecutive descending natural numbers.  n! = n(n-1)(n-2) ... 3•2•1 Let's solve,  6! = 6*5*4*3*2*1 = 720 /6  5! = 5*4*3*2"1 = 120 /5  4! = 4*3*2*1 = 24 /4  3! = 3*2*1 = 12 /3  2! = 2*1 = 2 /2  1! = 1 = 1 /1  0! = ? = 1 It follows pattern wherein, from larger number factorial divided by the number, 6!/6, which descends to 0!. From the pattern, 0! is obviously 1. Therefore, 0! = 1. Then, from the formula, n! = n(n-1) Wi…

Exploring the World of Permutations: A Guide to Arranging Objects

Permutations are a fundamental concept in mathematics that deals with the different ways to arrange objects in a specific order. Understanding permutations is crucial in various fields, including probability, statistics, computer science, and even everyday life scenarios. This article will delve into the different types of permutations, their formulas, and illustrate them with practical examples.   What are Permutations?   A permutation is an arrangement of objects in a specific order. The key difference between a permutation and a combin…

Understanding Probability

Probability is the branch of mathematics that quantifies uncertainty.   An experiment is any process with unpredictable results. Each repetition is a trial. Examples include flipping a coin or rolling a die.   An outcome is a single result of an experiment.   An event is a group of outcomes. A simple event consists of only one outcome.   The sample space encompasses all possible outcomes of an experiment. Sample Problems A. 1. Experiment: Rolling a die Sample Space (S): S= {1, 2, 3, 4, 5, 6}   Events (E): The event that even numbers ap…

Understanding Factorial Notation

Factorial notation is shorthand for multiplying consecutive descending natural numbers. n! = n(n-1)(n-2) ... 3•2•1 NOTE: 0! = 1 Sample Problems: A. 1. 3! = 2. 5!= 3. 7! = B. List down all the possible ways of writing: 1. 3! = 2. 5!= 3. 7! = A. Formula: n! = n(n- 1)! Solutions: 1. 3! =3•2•1 = 6 2. 5! =5•4•3•2•1= 120 3. 7! =7•6•5•4•3•2•1 = 5040 Equivalents B. 1. n! with n = 3 n! = n(n- 1)! 3! = 3(3 -1)! 3! = 3(2)! or 3! =3•2! or 3! =3•2•1! B. 2. n! with n = 5 n! = n(n- 1)! 5! = 5(5- 1)! 5! = 5(4)! or 5! =5•4•3! or 5! =5•4•3•2! or 3! =5•4•3•2•1! C.…

Combination

It is a mathematical technique that determines the possible arrangements in a collection of items where the order of the selection does not matter. In combination, you can select the items in any order. The number of combinations of a set of n distinct objects taken at a time, is given by nCr = n!/[(n-r)!r!] ; r =/< n Where:: C is the total number of COMBINATIONS n is the total number of distinct objects in the set r is the total number of choosing objects in the set Sample Problems: Solve the following combinations. 1. 9C5 2. 14C2 S…

Circular Permutation

The formula for calculating the number of ways to arrange n objects in a circle is: P= (n -1)! Where: P is the total number of permutations n is the total number of objects in the set Sample Problems 1. How many ways can five people be seated around a circular table? Formula:  P= (n -1)!  Given:  P =? n =5 Solution: P= (n-1)! P= (5-1)! P=4! =4-3-2 ·1 P= 24, ways 2. How many ways can seven cheerleaders be arranged in a circle? Formula:  P= (n-1)!  Given:  P =? n =7 Solution: P= (n- 1)! P= (7-1)! P=6! =6•5.4.3-2.1 P=720 ways

Distinguishable Permutation

The number of permutation of n objects where there are n 1  repeated items, n 2  repeated items, ... n k  repeated items. P = n!/n 1 !*n 2 !...n k ! Where: P is the total number of permutations n is the total number of objects in the set „n 1 , n 2 ,..n k  is the total number of repeated items/objects Sample Problems: Find the Permutation of the letters of the following: 1. MANILA 2. PHILIPPINES Solution: 1. MANILA Given: P = ?; n = 6; n 1  = 2 Formula: P = n!/(n 1 !*n 2 !*...n k !) Solution: P = 6!/2! = 6*5*4*3*2!/2! = 180 2. PHILIPPINES Given: P…

Permutation with Repetition and Replacement

The number of orderings of n objects taken r at a time, with repetition, is given by P=n r Where: P is the total number of permutations n is the total number of objects in the set r is the total number of choosing objects in the set Sample Problems: 1. How many (5-letter) ordered codes can be formed with the letters of the set (A, B, C & D) if we allow to use repeated use of the same letter? Given: P =?; n =4; r =5 Formula:  P=n r Solution: P= n r P= (4) 5  =4*4*4*4*4 (no need to write) P= 1024 ordered codes 2. A standard deck of card…

Permutation without Repetition

Before we proceed, bear in mind that you need the idea of factorial in this topic. A factorial is the product of all positive integers less than or equal to a given positive integer. We write this using an exclamation mark (!), for example, 5! = 5 × 4 × 3 × 2 × 1. A permutation is an ordered arrangement of objects. In other words, it's a way of arranging a set of things where the order matters. The number of permutations of n objects taken r at a time (written as nPr ) is calculated using a specific formula. For example, in how many…

Permutation and Combination

A permutation is an arrangement of objects in a specific order, while a combination is a selection of objects where order doesn't matter. Here's a simple way to think about it: - Permutation : Imagine you have 3 letters, A, B, and C. You want to know how many different ways you can arrange them. You have 6 possible permutations: ABC, ACB, BAC, BCA, CAB, CBA. - Combination : Now, imagine you have the same 3 letters, and you want to know how many different ways you can choose 2 letters. You have 3 possible combinations: AB, AC, BC. Notice that…

Combination

Permutations  count ordered lists. Think of a race—first place matters! But in many cases, order doesn't matter. For instance, in a card hand, the order you receive the cards is irrelevant. When order is unimportant, we use combinations instead of permutations. How to compute the number of Combinations for a given set of objects To compute the number of permutations for a given set of objects, you can use the formula: nCr = n!/r!(n-r)! where n is the total number of objects, r is the number of objects to arrange, and ! denotes factorial. For example…

Permutation

The number of ways to arrange items from a set is called a permutation. The formula for calculating permutations is P(n, r) = n!/(n-r)!, where n is the total number of items and r is the number of items selected. This formula uses factorials, and we'll practice calculating permutations using this formula. How to compute the number of permutations for a given set of objects To compute the number of permutations for a given set of objects, you can use the formula: nPr = n!/(n-r)! where n is the total number of objects, r is the number of obj…