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B.Sc Mathematics Sequences, Probability and Counting Theory — practice questions — page 3

Explore 92 free Sequences, Probability and Counting Theory questions in B.Sc Mathematics Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Sequences, Probability and Counting Theory questions and exercises with answers

Showing questions 61–90 of 92.

Nine swimmers race and the one named Ariel is known to have won first place. In how many ways can the first three places then be filled?Read the answer and worked solution →What is a permutation?Read the answer and worked solution →In the permutation formula P(n, r), why is n factorial divided by (n - r) factorial?Read the answer and worked solution →A club of six members must elect a president, a vice president and a treasurer. In how many ways can the three posts be filled?Read the answer and worked solution →What is a combination?Read the answer and worked solution →A meal comes with two side dishes chosen from five options, and the order of the sides does not matter. How many selections are possible?Read the answer and worked solution →Choosing 2 side dishes from 5 and choosing 3 side dishes from 5 both give 10. Which identity explains that?Read the answer and worked solution →A baked potato can take any number of four available toppings, including none of them. How many different orders are possible?Read the answer and worked solution →Why does a set of n distinct objects have exactly 2^n subsets?Read the answer and worked solution →How many distinct rearrangements are there of the letters in the word DISTINCT?Read the answer and worked solution →What is a binomial?Read the answer and worked solution →How is the binomial coefficient written as n over r calculated?Read the answer and worked solution →Can a binomial coefficient be a fraction?Read the answer and worked solution →Evaluate the binomial coefficient 9 over 7.Read the answer and worked solution →In the expansion of (x + y)^5, what do the exponents in any single term add up to?Read the answer and worked solution →In the expansion of (x + y)^5, what is the coefficient of the term in x^3 y^2?Read the answer and worked solution →How is each interior entry of Pascal's Triangle produced?Read the answer and worked solution →To find the tenth term of the expansion of (x + 2y)^16, which value of r goes into the formula for the (r + 1)th term?Read the answer and worked solution →In probability, what is an experiment?Read the answer and worked solution →What is the sample space of an experiment?Read the answer and worked solution →In a probability model, what is an event?Read the answer and worked solution →What does a probability of 0 indicate about an event?Read the answer and worked solution →What must be true of all the probabilities listed in a probability model?Read the answer and worked solution →A fair six-sided number cube is rolled. What is the probability of rolling an odd number?Read the answer and worked solution →When finding the probability that E or F occurs, why is the probability of E and F subtracted?Read the answer and worked solution →One card is drawn from a standard deck. What is the probability of drawing a heart or a 7?Read the answer and worked solution →What makes two events mutually exclusive?Read the answer and worked solution →One card is drawn from a standard deck. What is the probability of drawing a heart or a spade?Read the answer and worked solution →The probability that a horse wins a race is 1/9. What is the probability that it loses?Read the answer and worked solution →Two six-sided number cubes are rolled. What is the probability that the sum is greater than 3?Read the answer and worked solution →

Worked practice questions

Nine swimmers race and the one named Ariel is known to have won first place. In how many ways can the first three places then be filled?

  1. 56 different ways
  2. 72 different ways
  3. 64 different ways
  4. 63 different ways
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Correct answer: A. 56 different ways

1. First place is already decided, so it contributes 1 option rather than 9. 2. That leaves 8 swimmers who could take second place. 3. After second is filled, 7 remain for third. 4. Multiplying gives 1 times 8 times 7 = 56 ways. 5. Using 9 times 8 for the remaining places would give 72, which forgets that Ariel is no longer available lower down. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.5 Counting Principles_

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What is a permutation?

  1. A selection in which order does not matter
  2. A selection of objects in which order matters
  3. A selection made without replacing objects
  4. A selection that keeps every object unused
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Correct answer: B. A selection of objects in which order matters

1. A permutation is an ordering of objects, so rearranging them produces a different permutation. 2. Lining up for a photograph or awarding first, second and third place are permutation problems. 3. A selection where order does not matter is a combination instead. 4. Whether objects are replaced is a separate question, so that description does not define a permutation. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.5 Counting Principles_

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In the permutation formula P(n, r), why is n factorial divided by (n - r) factorial?

  1. To restore the objects left unchosen
  2. To remove the repeated identical items
  3. To cancel the items not being lined up
  4. To divide out the orderings of r items
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Correct answer: C. To cancel the items not being lined up

1. The quantity n! counts the orderings of all n objects at once. 2. Only r of them are actually being placed in order, so the remaining n - r are irrelevant. 3. Dividing by (n - r)! cancels the orderings of those unwanted items. 4. Dividing instead by r! would remove the orderings of the chosen items, and that is what turns a permutation count into a combination count. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.5 Counting Principles_

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