Combinatorics & Probability

1. Almost all counting problems can be thought of in terms of what?

The average of all options

The ratio of available choices

The sum of all possibilities

The number of spots that are available and the number of possibilities for each spot.

Counting problems involve considering the available spots and the possibilities for each spot, not just the sum, average, or ratio.

Explanation

Counting problems involve considering the available spots and the possibilities for each spot, not just the sum, average, or ratio.

2. What is the concept of permutations in mathematics?

Arrangement matters - the number of elements often corresponds to the number of available slots for these elements or where items cannot be reused

Permutations refer to combinations where the order does not matter.

The number of elements corresponds to the total sum of the elements involved.

Arrangements in permutations allow for the reuse of elements in different slots.

Permutations involve arranging items where the order matters and no item can be reused. It is different from combinations, which focus on selections where the order does not matter.

Explanation

Permutations involve arranging items where the order matters and no item can be reused. It is different from combinations, which focus on selections where the order does not matter.

3. What is a combination?

Involves repeated selection of elements from a pool

Order does not matter - often corresponds to when a smaller group of elements is drawn from a larger pool

A type of permutation

When elements are arranged in a specific order

A combination refers to a selection of elements where the order is not important, distinguishing it from permutations where order matters.

Explanation

A combination refers to a selection of elements where the order is not important, distinguishing it from permutations where order matters.

4. What is the value of zero factorial?

2

0

Equals 1

Undefined

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. As per mathematical convention, the value of 0 factorial is defined to be 1.

Explanation

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. As per mathematical convention, the value of 0 factorial is defined to be 1.

5. When determining a permutation where the number of available items (N) equals the number of spots (K) in which to arrange those items, what is the correct answer?

N*K

N^K

Simply N!

N

In permutations, when the number of items (N) equals the number of spots (K) to arrange those items, the formula to use is N!

Explanation

In permutations, when the number of items (N) equals the number of spots (K) to arrange those items, the formula to use is N!

6. For all permutations where N does not equal K, the total number of permutations is calculated as:

K! / (N-K)!

N! / K!

N^K

The total number of permutations = N! / (N-K)!

The correct formula for calculating the total number of permutations when N does not equal K is N! / (N-K)! which is derived from the principle of permutations without repetition. Options 1, 2, and 3 are incorrect as they do not follow the correct formula for this specific scenario.

Explanation

The correct formula for calculating the total number of permutations when N does not equal K is N! / (N-K)! which is derived from the principle of permutations without repetition. Options 1, 2, and 3 are incorrect as they do not follow the correct formula for this specific scenario.

7. If a question asks for the number of different arrangements of x elements when those elements are arranged in a circle, what formula should be used?

(x-1)!

X^2

X

X!

When arranging elements in a circle, one element can be fixed as a starting point, reducing the number of total arrangements by 1. Therefore, the correct formula is (x-1)!

Explanation

When arranging elements in a circle, one element can be fixed as a starting point, reducing the number of total arrangements by 1. Therefore, the correct formula is (x-1)!

8. The number of arrangements of N elements where certain elements are repeated is represented by

(N-A)!B!

N! / A!B! where N is the number of items being arranged and each factorial in the denominator represents an item that is repeated and the variable represents the number of times the items repeats ex: arranging the letters A, B and B so there are 3 items, N=3 and we have one item that repeats 2 tiems. Therefore, the answer 3!/2! = 3

N! / (A!*B!)

N! / A!*B!*C!

The correct formula to calculate the number of arrangements of N elements with repeated elements is N! divided by the factorial of the number of times each repeated element appears. Each factorial in the denominator corresponds to the repeated elements and their occurrences.

Explanation

The correct formula to calculate the number of arrangements of N elements with repeated elements is N! divided by the factorial of the number of times each repeated element appears. Each factorial in the denominator corresponds to the repeated elements and their occurrences.

9. If K is changing in a permutation problem, what should you do?

Count the number of permutations without considering the changing value of K.

Always set K to a fixed value to simplify the problem.

1) Look for trigger words like "At least", "At most", and "Or"2) If K is changing in the problem, sum up all the results for the different values of K given one value for N

Divide the sum of permutations by the changing value of K.

In a permutation problem where K is changing, it is essential to sum up all the results for the different values of K given one value for N to accurately account for all possible permutations.

Explanation

In a permutation problem where K is changing, it is essential to sum up all the results for the different values of K given one value for N to accurately account for all possible permutations.

10. What is the equation for solving most combinations?

N! * K! / (N - K)!

The number of unordered arrangements consisting of K items selected from a pool of N elements: N! / K! * (N - K)!

N! / (K! * (N - K))!

N! * (N - K)! / K!

The formula for combinations involves calculating the number of ways to choose K items from a pool of N elements without regard to the order. The correct formula is N! / K! * (N - K)!, which represents the number of unordered arrangements.

Explanation

The formula for combinations involves calculating the number of ways to choose K items from a pool of N elements without regard to the order. The correct formula is N! / K! * (N - K)!, which represents the number of unordered arrangements.

11. Examples of a permutation.

3) Random numbers picked from a hat without replacement.

1) Colors of a rainbow without any specific order.

1) Letters in a password without repetition2) Arrangements of 5 students in 5 seats

2) Different types of animals in a zoo.

Permutations involve arranging items in a specific order or sequence without repetition. The correct examples provided involve arranging objects (letters in a password, students in seats) in a specific order - making them permutations. Incorrect answers such as colors of a rainbow, types of animals, or random numbers do not adhere to the concept of permutations.

Explanation

Permutations involve arranging items in a specific order or sequence without repetition. The correct examples provided involve arranging objects (letters in a password, students in seats) in a specific order - making them permutations. Incorrect answers such as colors of a rainbow, types of animals, or random numbers do not adhere to the concept of permutations.

12. Examples of a permutation with repeating elements.

2) Rearranging a deck of cards without any duplicates

1) Sorting a list of unique numbers

3) Organizing a set of distinct colors

1) Arrangements of different vases, two of which are identical2) Arrangements of letters, some of which are repeated

Permutations with repeating elements involve arranging items where some elements are identical, leading to multiple possible variations.

Explanation

Permutations with repeating elements involve arranging items where some elements are identical, leading to multiple possible variations.

13. Which of the following is an example of a combination?

2) Employees in a company

3) Players on a soccer team

4) Actors in a movie

1) Members or a committee or students in a study group

A combination involves selecting a group of items or individuals where the order does not matter. In the correct answer, members of a committee or students in a study group can be combined in different ways without changing the outcome, making it an example of a combination.

Explanation

A combination involves selecting a group of items or individuals where the order does not matter. In the correct answer, members of a committee or students in a study group can be combined in different ways without changing the outcome, making it an example of a combination.

14. The entire spectrum of probabilities of any event falls between.

-1 to 1, inclusive

0 and 1, inclusive

0 to 100, inclusive

1 to 10, inclusive

Probabilities range from 0 (impossible) to 1 (certain), including all values in between. Therefore, the correct answer is 0 and 1, inclusive.

Explanation

Probabilities range from 0 (impossible) to 1 (certain), including all values in between. Therefore, the correct answer is 0 and 1, inclusive.

15. The probability of a single event is determined by:

The time of day when the event happens.

The luck of the person involved.

The ratio of the outcomes when this event occurs to the total number of possible outcomes.

The color of the event's outcome.

Probability is based on the ratio of specific outcomes to the total possible outcomes, not on luck, color, or time of day.

Explanation

Probability is based on the ratio of specific outcomes to the total possible outcomes, not on luck, color, or time of day.

16. What is the formula to calculate the probability of an outcome?

Probability = Number of outcomes when A occurs - Total number of possible outcomes

Probability = Total number of possible outcomes / Number of outcomes when A occurs

# of outcomes when A occurs / # of possible outcomes

Probability = Number of outcomes when A occurs * Total number of possible outcomes

The correct formula to calculate probability is by dividing the number of outcomes when A occurs by the total number of possible outcomes. This ratio gives the likelihood of event A happening in relation to all possible outcomes.

Explanation

The correct formula to calculate probability is by dividing the number of outcomes when A occurs by the total number of possible outcomes. This ratio gives the likelihood of event A happening in relation to all possible outcomes.

17. When determining the probability of multiple independent events, what is the correct method?

Take the average of the individual probabilities.

Simply multiply the individual probabilities.

Add the individual probabilities together.

Divide the individual probabilities by each other.

When events are independent, the correct method to determine the overall probability is to multiply the individual probabilities of each event. This is based on the principle of the multiplication rule for independent events in probability theory.

Explanation

When events are independent, the correct method to determine the overall probability is to multiply the individual probabilities of each event. This is based on the principle of the multiplication rule for independent events in probability theory.

18. When determining the probability of multiple dependent events,

Multiply the probability of the first event times the probability of the second event assuming the first event has occured

Divide the probability of the first event by the probability of the second event.

Ignore the probability of the first event and only consider the probability of the second event.

Add the probability of the first event to the probability of the second event.

When dealing with dependent events, the probability of both events occurring is found by multiplying their individual probabilities given that one event has already occurred.

Explanation

When dealing with dependent events, the probability of both events occurring is found by multiplying their individual probabilities given that one event has already occurred.

19. The probability of dependent events occurring together is calculated using which formula?

P(A and B) = P(A) + P(B) - P(A * B): where P(A and B) is the probability of both events occurring together

P(A and B) = P(A) * P(B): where P(A and B) is the probability that either event A or B will occur

P(A and B) = P(A) * Pa(B): where P(A and B) is the probability that both these events will occur; P(A) is the probability of event A; Pa(B) is the probability of that event B will occur assuming that A has already occured

P(A and B) = P(A) + P(B): where P(A and B) is the probability of either event A or event B occurring together

20. When calculating the probability of two things happening simultaneously in probability, what method should be used?

One at a time, without replacement

Solely based on the outcome of the first event

Together, with replacement

Randomly, regardless of order

When calculating the probability of two events happening simultaneously, they should be done one at a time without replacement to accurately determine the combined probability.

Explanation

When calculating the probability of two events happening simultaneously, they should be done one at a time without replacement to accurately determine the combined probability.

21. What is the correct formula for calculating the probability of either event A or event B occurring?

P(A) * P(B)

P(A) - P(B)

P(A) + P(B)

On probability questions AND means multiply; OR means add (and subtract if necessary)

22. When calculating the probability of 'at least one' occurrence, what is the recommended approach?

Skip calculating the probability and focus on other aspects of the problem.

It is almost alwyas easiet to find the probability of none which is complementary event to at least one

Assume the probability of 'at least one' is always 100% without further analysis.

Try to determine the probability of 'at least one' directly without considering the complementary event.

When dealing with 'at least one' probabilities, focusing on the complementary event of 'none' provides a simpler and more straightforward calculation method.

Explanation

When dealing with 'at least one' probabilities, focusing on the complementary event of 'none' provides a simpler and more straightforward calculation method.

23. In a coin flip, the probability of any series of n events must be what?

1/n

(1/2)^n

2^n

N/2

The probability of any series of n events in a coin flip is calculated by raising 1/2 to the power of n. This is because there are two possible outcomes (heads or tails) for each coin flip, each with a probability of 1/2.

Explanation

The probability of any series of n events in a coin flip is calculated by raising 1/2 to the power of n. This is because there are two possible outcomes (heads or tails) for each coin flip, each with a probability of 1/2.

24. What marks the completion of a certain event in the realm of chance?

Infinite Possibilities

Beginning of Certainty

Uncertain Outcome

End of Probability

The correct answer 'End of Probability' signifies the definitive end point of the probabilistic outcomes, indicating that the event has been resolved and the likelihood of different outcomes no longer exists.

Explanation

The correct answer 'End of Probability' signifies the definitive end point of the probabilistic outcomes, indicating that the event has been resolved and the likelihood of different outcomes no longer exists.