In mathematics, the concept of combinations helps in identifying the number of possible selections from a larger set where the sequence of selection is not significant. The formula nCr = n! / r!(n-r)! is typically used to solve these problems. Here, 'n' represents the total items, 'r' stands for the items chosen, and '!' signifies factorial.
| Topic | Problem | Solution |
|---|---|---|
| None | Results of a survey of fifty students indicate th… | Let the total number of students be represented as \(total\_students = 50\) |
| None | Suppose you have some money to invest-for simplic… | Given that the return on investment is given by the equation \(R=w R_{s}+(1-w) R_{b}\), where \(R_{… |
| None | 38. How many ways can you order a sandwich if you… | This is a problem of combinations. Each ingredient (ham, mustard, mayo, cheese) can be either inclu… |
| None | You volunteer to help drive children at a charity… | This is a combination problem. We are choosing 7 children out of 17, without regard to the order in… |
| None | The manufacturer of a fertilizer guarantees that,… | This problem is a binomial probability problem. The probability of success, which is the germinatio… |
| None | How many different ways can a(n) 12-member jury b… | This problem is about selecting a 12-member jury from 24 possible jury members. The order in which … |
| None | A poll is given, showing $20 \%$ are in favor of … | This problem is a binomial probability problem. The binomial distribution model deals with finding … |
| None | Based on a poll, $50 \%$ of adults believe in rei… | This problem is a binomial probability problem. We have a binomial experiment where each trial (adu… |
| None | Assume that when adults with smartphones are rand… | Define the problem as a binomial probability problem, where 'n' is the number of trials (8 adult sm… |
| None | The Smith family was one of the first to come to … | The problem is asking for the probability of having at least 2 girls and at most 4 girls in a famil… |
| None | To win at LOTTO in a certain state, one must corr… | This problem is about calculating the number of combinations. In combinatorics, a combination is a … |
| None | Suppose you have 16 tubes of paint. How many dist… | We are given 16 tubes of paint and asked to find the number of distinct color groupings we can make… |
| None | Clarice, Dominique, John, and Marco work for a pu… | The sample space of the experiment is all possible combinations of 2 employees out of 4. This can b… |
| None | If a single card is drawn from a standard 52-card… | A standard deck of 52 cards contains 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13… |
| None | How many ways can Marie choose 3 pizza toppings f… | This problem is about combinations. In combinatorics, a combination is a selection of items without… |
| None | In order to conduct an experiment, 4 subjects are… | This problem is about combinations. We are choosing 4 subjects from a group of 41 without regard to… |
| None | To win at LOTTO in one state, one must correctly … | We are given a LOTTO game where one must correctly select 7 numbers from a collection of 52 numbers… |
| None | A standard 52-card deck contains four queens, twe… | A standard 52-card deck contains four queens, twelve face cards, thirteen hearts (all red), thirtee… |
| None | A standard 52-card deck contains four queens, twe… | A standard 52-card deck contains four queens, twelve face cards, thirteen hearts (all red), thirtee… |
| None | A standard 52-card deck contains four queens, twe… | A standard 52-card deck contains four queens, twelve face cards, thirteen hearts (all red), thirtee… |
| None | How many ways can a male and a female be selected… | We need to select one male and one female from the club. The number of ways to select one male from… |
| None | An electronics store receives a shipment of 19 gr… | Translate the problem into English: An electronics store receives a shipment of 19 graphing calcula… |
| None | $-5 \cdot$ (3 marks) A lock has 5 dials, each wit… | Find the total number of possible sequences: \(10^5 = 100000\) |
| None | Now consider a larger group of students: Alice, B… | Break the problem into two cases: (1) Lisa and Claire are in the same group, but Frank is not. (2) … |
| None | Question 6 of 29 There are 8 seniors on the high … | This is a combination problem because the order of the co-captains does not matter. We need to find… |
| None | Roll two dice virtually (by calculator: [APP] 10.… | Roll the two dice virtually for \( n \) times, where \( n = 36, 50, 100, 1000, 10000 \). Record the… |
| None | Watch help video If a fair coin is tossed 5 times… | Use the binomial probability formula: \( P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k} \) |