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Articles by Eva Sharma
17 articles
Permutations and Combinations (Concept, Examples, C++ Program)
Permutations and Combinations refer to the arrangements of objects in mathematics. Permutation − In permutation, the order matters. Hence, the arrangement of the objects in a particular order is called a permutation. Permutations are of two types − Permutation with repetition Suppose we have to make a three-digit code. Some possible numbers are 123, 897, 557, 333, 000, and 001. So how many numbers can we make like this? Let us look at it this way− In the once place, we have ten options − 0-9 Similarly, at the tenth and the hundredth place also, we have ten options. 0-9. ...
Read MoreHCF of an array of fractions (or rational numbers)
HCF or the Highest common factor of two or more numbers refers to the highest number which divides them. A rational number is the quotient p/q of two numbers such that q is not equal to 0. Problem Statement Given an array with fractional numbers, find the HCF of the numbers. Example 1 Input [{4, 5}, {10, 12}, {24, 16}, {22, 13}] Output {2, 3120} Explanation The fractional numbers given are: 4/5, 10/12, 24/16 and 22/13 2/3120 is the largest number that divides all of them. Example 2 Input [{18, 20}, {15, 12}, {27, 12}, {20, 6}] ...
Read MoreStella Octangula Number
In mathematics, a Stella Octangula number is a figurate number based on the Stella Octangula, of the form n(2n2 − 1). Stella Octangula numbers which are perfect squares are 1 and 9653449. Problem Statement Given a number n, check whether it is the Stella Octangula number or not. The sequence of Stella Octangula numbers is 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990 Example1 Input x = 14 Output Yes Explanation $$\mathrm{For\: n = 2, expression \:n\lgroup 2n^2 – 1\rgroup is\: 14}$$ Example2 Input n = 22 Output No Explanation $$\mathrm{There \:is\: no\: ...
Read MoreNicomachus’ Theorem
According to Nicomachus’ Theorem, the sum of the cubes of the first n integers is equal to the square of the nth triangular number. Or, we can also say − The sum of cubes of first n natural numbers is equal to square of sum of first natural numbers. Putting it algebraically, $$\mathrm{\displaystyle\sum\limits_{i=0}^n i^3=\lgroup \frac{n^2+n}{2}\rgroup^2}$$ Theorem $$1^3 = 1$$ $$2^3 = 3 + 5$$ $$3^3 = 7 + 9 + 11$$ $$4^3 = 13 + 15 + 17 + 19\vdots$$ Generalizing $$n^3 =\lgroup n^2−n+1\rgroup+\lgroup n^2−n+3\rgroup+⋯+\lgroup n^2+n−1\rgroup$$ Proof By Induction For all n Ε Natural ...
Read MoreEqual sum array partition excluding a given element
Problem Statement For an index and an arr[]. Check if the array[] can be partitioned into two disjoint sets, excluding arr[index] such that the sum of both sets has equal value. Example 1 Input arr[] = {4, 3, 1, 2}, Index = 1 Output No Explanation We have to exclude arr[1] = 3 All possible sets are − Set 1: (4), Set 2: (2, 1), sum = 4≠3 Set 1: (4, 1), Set 2: (2), sum = 5≠2 Set 1: (4, 2), Set 2: (1), sum = 6≠1 No combination satisfies the conditions. Example 2Input arr[] ...
Read MoreSquared Triangular Number (Sum of Cubes)
A square triangular number, also referred to as a triangular square number, is a number that is both a triangular number and a perfect square. Square triangular numbers have an unlimited number of possible values; the first few are − 0, 1, 36, 1225, 41616... A triangular number or triangle number counts objects arranged in an equilateral triangle. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, ...
Read MoreThe time when the minute hand and the hour hand coincide after a given hour
When the minute hand moves from 12 to 12 in one hour, the hour hand also moves from the previous hour to the next. Hence, every hour, the minute hand and the hour hand coincide once. Problem Statement Given an input hour, find the time in minutes when the hour hand and the minute hand coincide within the next hour. Examples Input − Hour = 4 Output − Coinciding time: 240/11 minutes. We will discuss the explanation further with the approach. Input − Hour = 5 Output − Coinciding time: 300/11 minutes. Explanation and the Approach ...
Read MoreSum of the series 5+55+555+.. up to n terms
5, 55, 555, ... is a series that can be derived from geometric progression and, thus, computed with the help of GP formulae. Geometric progression is a type of series in which each succeeding term is the product of some specific term (ratio) with the preceding term. We will utilize the knowledge of GP, to find the sum of the given series. Problem Statement Given a number n, find the sum of the series 5+5+555+... up to n terms. Examples Input − N = 3 Output − 595 Explanation 5 + 5 + 555 = 595. ...
Read MoreSum of products of all combinations taken (1 to n) at a time
There can be multiple combinations of numbers if taken 1 to n at a time. For example, if we take one number at a time, the number of combinations will be nC1. If we take two numbers at a time, the number of combinations will be nC2. Hence, the total number of combinations will be nC1 + nC2 +… + nCn. To find the sum of all combinations, we will have to use an efficient approach. Otherwise, the time and space complexities will go very high. Problem Statement Find the sum of products of all the combinations of numbers taken ...
Read MoreSort on the basis of number of factors using STL
Sorting a vector using STL is a piece of cake. We can use the famous sort() function to perform the task. The real challenge is to count the number of factors for each number. A factor is a number which divides another number completely, i.e. with zero remainder. Traversing through all the numbers to count the factors might be an approach but we will try to optimize and reach efficient solutions in this article. Problem Statement Sort a given array based on the number of factors of each number in increasing order. Thus, the number having the lowest number of ...
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