Counting & Permutations — Practice Set

Bank code: STA-PRM · Section: statistics · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Ask whether order matters, draw one slot per position and multiply the choices, then divide by the factorial of every repeat.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~3 min

warm-up — these must be automatic

Q1 · STA-PRM-001 · MCQ · 40s

Five distinguishable books are placed side by side in a row on a shelf, each book used exactly once. Two placements count as different if the books appear in a different left-to-right order. How many placements are possible?

(A) 15
(B) 20
(C) 60
(D) 120
(E) 3125

Q2 · STA-PRM-003 · Numeric Entry · 35s

What is the value of 0! + 1! + 2! + 3! ? Enter your answer as a number.

Numeric entry — write the number.

Q3 · STA-PRM-004 · MCQ · 50s

A club must fill the three distinct offices of president, vice president and treasurer from its 8 members. No member may hold more than one office. In how many different ways can the three offices be filled?

(A) 24
(B) 56
(C) 336
(D) 512
(E) 40320

Q4 · STA-PRM-010 · QC · 45s

Three books, A, B and C, are distinguishable.

Column A: The number of different left-to-right orders in which all 3 books can be placed in a row Column B: The number of different 3-book selections that can be made from these 3 books, where the order within a selection does not matter

Q5 · STA-PRM-013 · MCQ · 35s

A lock code consists of two characters. The first character is chosen from the set {1, 2, 3} and the second character is chosen from the set {4, 5}. The two choices are made independently. How many different codes are possible?

(A) 2
(B) 3
(C) 5
(D) 6
(E) 9

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · STA-PRM-025 · MCQ · 80s

Six distinguishable people are to be seated in the six seats of a circular table. Two seatings are considered the same if one is a rotation of the other; reflections are considered different. In how many ways can they be seated if Anna and Ben must occupy adjacent seats?

(A) 24
(B) 48
(C) 120
(D) 240
(E) 720

Q7 · STA-PRM-027 · MCQ · 70s

How many distinct orderings are there of the six letters of PEPPER, that is, of the multiset P, P, P, E, E, R? Letters that are the same are indistinguishable from one another.

(A) 20
(B) 60
(C) 120
(D) 360
(E) 720

Q8 · STA-PRM-028 · Numeric Entry · 80s

Three of the six letters A, B, C, D, E, F are written in a row to form a 3-letter arrangement. No letter may be used more than once, and two arrangements that use the same three letters in different orders count as different. How many of these arrangements contain the letter A? Enter your answer as a number.

Numeric entry — write the number.

Q9 · STA-PRM-026 · QC · 80s

n is an integer and n >= 3.

Column A: P(n, 3), the number of ordered arrangements of 3 objects chosen from n distinct objects Column B: n^3 - n

Q10 · STA-PRM-030 · QC · 65s

Column A: The number of distinct orderings of the four letters of NOON, that is, of the multiset N, N, O, O Column B: The number of distinct orderings of the four letters of MOON, that is, of the multiset M, O, O, N

Q11 · STA-PRM-029 · Select all that apply · 85s

The five distinct letters A, B, C, D, E are written in a row, each used exactly once, so there are 5! = 120 arrangements in all. Which of the following expressions give the number of these arrangements that begin with A? Select all that apply.

(A) 24
(B) P(4, 4)
(C) P(5, 4)
(D) 5!/5
(E) 5! - 4!

Q12 · STA-PRM-036 · Select all that apply · 90s

Each option below names a 5-letter string and gives the multiplicity of each of its letters; letters that are the same are indistinguishable. For which strings is the number of distinct orderings of the five letters strictly greater than 30? Select all that apply.

(A) TIGER (T, I, G, E, R, all different)
(B) TEETH (T x 2, E x 2, H)
(C) ELITE (E x 2, L, I, T)
(D) AABBB (A x 2, B x 3)
(E) SEVEN (E x 2, S, V, N)

Q13 · STA-PRM-035 · MCQ · 75s

Five flags of five different colours, one of which is red and one of which is blue, are hung in a row on five hooks, one flag per hook. In how many of the possible arrangements does the red flag hang somewhere to the left of the blue flag? The two flags need not be adjacent.

(A) 24
(B) 48
(C) 60
(D) 72
(E) 120

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · STA-PRM-039 · MCQ · 100s

Three men and three women, all six people distinguishable, are to be seated in a row of six chairs. In how many of the arrangements is no man seated immediately next to another man?

(A) 24
(B) 36
(C) 72
(D) 144
(E) 720

Q15 · STA-PRM-040 · Numeric Entry · 105s

How many distinct orderings of the eleven letters of MISSISSIPPI, that is, of the multiset M, I, I, I, I, S, S, S, S, P, P, begin with S and end with S? Enter your answer as a number.

Numeric entry — write the number.

Q16 · STA-PRM-041 · MCQ · 100s

A firm has 8 distinguishable employees. First, 3 of them are chosen to form a safety committee; the three members have equal standing, so the order in which they are named does not matter. Then, from the 5 employees not on the committee, 2 are chosen to hold the two distinct titles of Floor Warden and Deputy Warden. In how many ways can the whole assignment be made?

(A) 76
(B) 560
(C) 1120
(D) 3360
(E) 6720

Q17 · STA-PRM-042 · QC · 95s

A and B are integers with A > B > 1.

Column A: P(A, B) Column B: A x P(A - 1, B - 1)

Q18 · STA-PRM-043 · MCQ · 100s

How many distinct orderings of the seven letters of ARRANGE, that is, of the multiset A, A, R, R, N, G, E, have the two A's NOT adjacent?

(A) 360
(B) 630
(C) 900
(D) 1260
(E) 2520

Q19 · STA-PRM-044 · Numeric Entry · 110s

How many 5-digit positive integers have five different digits and are divisible by 5? (A 5-digit integer cannot begin with 0.) Enter your answer as a number.

Numeric entry — write the number.

Q20 · STA-PRM-045 · Select all that apply · 110s

Eight distinguishable people are seated in the eight seats of a circular table. Two seatings are considered the same if one is a rotation of the other; reflections are considered different. Which of the following statements are true? Select all that apply.

(A) The total number of seatings equals 8!/7
(B) The total number of seatings is 7! = 5040
(C) If two particular people must sit directly opposite each other, the number of seatings is 2 x 6! = 1440
(D) If two particular people must sit in adjacent seats, the number of seatings is 2 x 6! = 1440
(E) If three particular people must occupy three consecutive seats, the number of seatings is 5! x 3! = 720

Q21 · STA-PRM-046 · MCQ · 95s

How many distinct orderings of the eight letters of PARALLEL, that is, of the multiset P, A, A, R, L, L, L, E, have the three L's all together in one consecutive block?

(A) 60
(B) 360
(C) 720
(D) 2160
(E) 3360

Q22 · STA-PRM-047 · QC · 90s

A certain string consists of 6 letters, which need not be all different.

Column A: The number of distinct orderings of the 6 letters of the string Column B: 360

Q23 · STA-PRM-048 · Numeric Entry · 100s

Ten horses, among them Thunder and Lightning, run a race in which there are no ties. In how many ways can the first three finishing positions (1st, 2nd and 3rd) be filled if Thunder finishes in the top three and Lightning does not? Enter your answer as a number.

Numeric entry — write the number.

Q24 · STA-PRM-050 · MCQ · 105s

Four boys and four girls, all eight people distinguishable, are to be seated in the eight seats of a circular table so that the genders alternate all the way around. Two seatings are considered the same if one is a rotation of the other; reflections are considered different. In how many ways can this be done?

(A) 24
(B) 144
(C) 288
(D) 576
(E) 5040

Q25 · STA-PRM-061 · Select all that apply · 105s

A 4-character code is formed by writing four digits in a row, each chosen from 0 through 9; the code may begin with 0, and the four positions are distinguishable. Which of the following statements are true? Select all that apply.

(A) If digits may be repeated, exactly 540 of the codes use two different digits, each appearing exactly twice
(B) If digits may be repeated, there are 10,000 possible codes
(C) If no digit may be repeated, there are 5,040 possible codes
(D) If digits may be repeated, exactly 4,960 of the codes use some digit more than once
(E) If digits may be repeated, exactly 40 of the codes use the same digit in all four positions

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · STA-PRM-054 · Numeric Entry · 125s

In how many distinct orderings of the ten letters of STATISTICS, that is, of the multiset S, S, S, T, T, T, A, I, I, C, do all three S's come before all three T's? Enter your answer as a number.

Numeric entry — write the number.

Q27 · STA-PRM-055 · Numeric Entry · 130s

Five married couples are to be seated in the ten equally spaced seats of a circular table so that each husband sits directly opposite his own wife. All ten people are distinguishable. Two seatings are considered the same if one is a rotation of the other; reflections are considered different. In how many ways can this be done? Enter your answer as a number.

Numeric entry — write the number.

Q28 · STA-PRM-056 · QC · 120s

Ten distinguishable beads are strung on a loop to make a necklace. Two necklaces are considered identical if one can be obtained from the other by rotating the loop or by flipping the loop over.

Column A: The number of distinct necklaces Column B: 10!/20

Q29 · STA-PRM-058 · Select all that apply · 125s

The seven letters of ALGEBRA, that is, the multiset A, A, L, G, E, B, R, are arranged in a row; the two A's are indistinguishable from each other. Which of the following statements are true? Select all that apply.

(A) The total number of distinct arrangements is 2520
(B) The number of arrangements that begin with A and end with A is 240
(C) The number of arrangements in which the two A's are adjacent is 720
(D) The number of arrangements in which the two A's are not adjacent is 1800
(E) The number of arrangements in which B occupies the 4th position is 720

Q30 · STA-PRM-060 · MCQ · 130s

A 4-person committee is to be chosen from 5 men and 4 women, and the four chosen people are then assigned the four distinct offices of Chair, Vice-Chair, Treasurer and Secretary, one office each. The committee must contain at least 2 women. How many different office assignments are possible?

(A) 81
(B) 486
(C) 1440
(D) 1944
(E) 3024

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