Worksheet: Counting Using Permutations

In this worksheet, we will learn how to use permutations to solve counting problems.

Q1:

In how many ways can 11 books be arranged on a shelf?

Q2:

Which of the following represents the number of ways in which 2 boys can sit in a row of 24 seats?

  • A ๏Šจ ๏Šจ ๐‘ƒ
  • B ๏Šจ ๏Šช ๏Šจ ๏Šช ๐‘ƒ
  • C ๏Šจ ๏Šซ ๏Šจ ๐‘ƒ
  • D ๏Šจ ๏Šช ๏Šจ ๐‘ƒ

Q3:

Which of the following represents the number of ways in which the letters of the word โ€œmelonsโ€ can be rearranged?

  • A 6 1 ๐‘ƒ
  • B 1 2 6 ๐‘ƒ
  • C 7 5 ๐‘ƒ
  • D 6 6 ๐‘ƒ

Q4:

Which of the following represents the number of ways we can form a password of length 13 characters using different English letters.

  • A ๏Šง ๏Šฉ ๏Šง ๏Šฉ ๐‘ƒ
  • B ๏Šจ ๏Šฌ ๏Šง ๏Šช ๐‘ƒ
  • C ๏Šง ๏Šฉ ๏Šง ๐‘ƒ
  • D ๏Šจ ๏Šฌ ๏Šง ๏Šฉ ๐‘ƒ

Q5:

Which of the following represents the number of ways in which a president and a vice president can be elected from a committee of 17 members?

  • A ๏Šง ๏Šญ ๏Šง ๏Šญ ๐‘ƒ
  • B ๏Šง ๏Šฏ ๏Šจ ๐‘ƒ
  • C ๏Šง ๏Šญ ๏Šง ๐‘ƒ
  • D ๏Šง ๏Šญ ๏Šจ ๐‘ƒ

Q6:

In how many ways can a three-digit number, with no repeated digits, be formed using the numbers 2, 9, and 8?

Q7:

Which of the following represents the number of ways a 4-digit number can be formed from 5 digits, given that each digit can NOT be used more than once?

  • A 4 4 ๐‘ƒ
  • B 9 4 ๐‘ƒ
  • C 6 4 ๐‘ƒ
  • D 5 4 ๐‘ƒ

Q8:

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestraโ€™s 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?

  • A 5 2 3 :
  • B 3 1 8 2 :
  • C 1 1 2 0 :
  • D 3 5 2 :
  • E 1 2 0 1 :

Q9:

Luna, Peter, and Johannsson are playing a game, where one of them needs to be a sheriff and one needs to be an outlaw. They write each of their names on a piece of paper and place them in a bowl. If two names are picked at random where the first will be a sheriff and the second will be an outlaw, how many different combinations are there?

Q10:

A company labels their products with codes that start with three English letters followed by eight non-zero digits. Which of the following represents the number of codes that can be created with no repetition of any letter or digit?

  • A ๏Šฉ ๏Šฉ ๏Šฎ ๏Šฎ ๐‘ƒ ร— ๐‘ƒ
  • B ๏Šจ ๏Šฌ ๏Šฉ ๏Šฏ ๏Šฎ ๐‘ƒ + ๐‘ƒ
  • C ๏Šฉ ๏Šฉ ๏Šฎ ๏Šฎ ๐‘ƒ + ๐‘ƒ
  • D ๏Šจ ๏Šฌ ๏Šฉ ๏Šฏ ๏Šฎ ๐‘ƒ ร— ๐‘ƒ

Q11:

A shopping centre has 6 doors that are both entrances and exits. Which of the following represents the number of ways you can enter and leave the shopping centre if you do NOT use the same door twice?

  • A ๏Šฌ ๏Šฌ ๐‘ƒ
  • B ๏Šฎ ๏Šจ ๐‘ƒ
  • C ๏Šฌ ๏Šง ๐‘ƒ
  • D ๏Šฌ ๏Šจ ๐‘ƒ

Q12:

Let ๐‘‹ = { ๐‘ฅ โˆถ ๐‘ฅ โˆˆ โ„ค , โˆ’ 1 6 โ‰ค ๐‘ฅ < 2 5 } and ๐‘Œ = { ( ๐‘Ž , ๐‘ , ๐‘ ) โˆถ ๐‘Ž , ๐‘ , ๐‘ โˆˆ ๐‘‹ , where ๐‘Ž , ๐‘ , ๐‘ } a r e d i s t i n c t e l e m e n t s . Which of the following represents the number of elements in ๐‘Œ ?

  • A ๏Šช ๏Šจ ๏Šฉ ๐‘ƒ
  • B ๏Šช ๏Šง ๏Šฉ ๏Šฎ ๐‘ƒ
  • C ๏Šช ๏Šจ ๏Šฉ ๏Šฏ ๐‘ƒ
  • D ๏Šช ๏Šง ๏Šฉ ๐‘ƒ

Q13:

Find the solution set of 4 2 ๐‘ƒ = ๐‘ƒ ๏— ๏Šฐ ๏Šฉ ๏Šฉ ๏— ๏Šฐ ๏Šซ ๏Šซ .

  • A { 4 }
  • B { 1 1 }
  • C { 2 2 }
  • D { 2 }

Q14:

Evaluate ๏Šง ๏Šจ ๏Šฉ ๏Šฉ ๐‘ƒ .

  • A 1 2 3 ร— 1 2 4 ร— 1 2 5
  • B 1 2 3 ร— 3
  • C 1 2 3 ร— 1 2 2 ร— 3
  • D 1 2 3 ร— 1 2 2 ร— 1 2 1

Q15:

If ๏— ๏Šฑ ๏˜ ๏Šจ ๐‘ƒ = 1 2 and ๏— ๏Šฐ ๏˜ ๏Šซ ๐‘ƒ = 6 , 7 2 0 , find ๏— ๏˜ ๐‘ƒ .

Q16:

Given that six times the number of permutations of three elements taken from a set of ๐‘› elements equals eight times the number of permutations of three elements taken from a set of ( ๐‘› โˆ’ 1 ) elements, find the value of ๐‘› .

Q17:

Calculate 1 6 โ‹… 1 5 โ‹… 1 4 .

Q18:

Calculate ๐‘› ๐‘Ÿ ๐‘› โˆ’ 1 ๐‘Ÿ โˆ’ 1 ๐‘ƒ โˆถ ๐‘ƒ .

  • A ๐‘› ๐‘Ÿ
  • B ๐‘Ÿ
  • C ๐‘Ÿ ๐‘›
  • D ๐‘›

Q19:

By using ๏Šจ ๏Šฉ ๏Ž ๐‘ƒ = 5 0 6 to find the value of ๐‘Ÿ , evaluate the expression ๏Šฉ ๏Ž ๏Šฐ ๏Šฌ ๏Šฉ ๐‘ƒ .

Q20:

Calculate ๐‘› ๐‘Ÿ ๐‘› ๐‘Ÿ โˆ’ 1 ๐‘ƒ รท ๐‘ƒ .

  • A ๐‘› โˆ’ ๐‘Ÿ
  • B ๐‘› โˆ’ ๐‘Ÿ โˆ’ 1
  • C ๐‘› + ๐‘Ÿ
  • D ๐‘› โˆ’ ๐‘Ÿ + 1

Q21:

If ๏Š ๏Šช ๏Š ๏Šฑ ๏Šง ๏Šฉ ๐‘ƒ = 7 ร— ๐‘ƒ , find ๏Š ๏Šฐ ๏Šฉ ๏Š ๏Šฑ ๏Šฉ ๐‘ƒ .

Q22:

If ๏— ๏Šฑ ๏˜ ๏Šช ๐‘ƒ = 7 3 , 4 4 0 and ๏— ๏Šฐ ๏˜ ๏Šง ๐‘ƒ = 2 6 , find ๏— ๏˜ ๐‘ƒ .

Q23:

Evaluate the expression ๏Š ๏Šฐ ๏Šช ๏Š ๏Šฑ ๏Šช ๐‘ƒ , given that ๏Š ๏Šฐ ๏Šซ ๏Šญ ๏Š ๏Šฐ ๏Šช ๏Šฌ ๐‘ƒ = 9 ร— ๐‘ƒ .

Q24:

In horse racing, a โ€œtrifectaโ€ occurs when a bettor wins by selecting the first three finishers in their exact order: 1st place, 2nd place, and 3rd place. How many different trifectas are possible if there are 14 horses in a race?

Q25:

If some set ๐ด has 7 elements, how many permutations does ๐ด have?

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