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๏Šจ๐‘ƒ
  • B๏Šจ๏Šช๏Šจ๏Šช๐‘ƒ
  • C๏Šจ๏Šช๏Šจ๐‘ƒ
  • D๏Šจ๏Šซ๏Šจ๐‘ƒ

Q3:

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

  • A๏Šฌ๏Šง๐‘ƒ
  • B๏Šง๏Šจ๏Šฌ๐‘ƒ
  • C๏Šฌ๏Šฌ๐‘ƒ
  • D๏Šญ๏Šซ๐‘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๏Šง๐‘ƒ
  • B๏Šจ๏Šฌ๏Šง๏Šฉ๐‘ƒ
  • C๏Šจ๏Šฌ๏Šง๏Šช๐‘ƒ
  • D๏Šง๏Šฉ๏Šง๏Šฉ๐‘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๏Šญ๐‘ƒ
  • B๏Šง๏Šญ๏Šง๐‘ƒ
  • C๏Šง๏Šฏ๏Šจ๐‘ƒ
  • D๏Šง๏Šญ๏Šจ๐‘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๏Šซ๏Šช๐‘ƒ
  • B๏Šฌ๏Šช๐‘ƒ
  • C๏Šช๏Šช๐‘ƒ
  • D๏Šฏ๏Šช๐‘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?

  • A3182:
  • B1201:
  • C523:
  • D1120:
  • E352:

Q9:

Michel, Kayla, and ChiWei 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๏Šฎ๐‘ƒ+๐‘ƒ
  • B๏Šจ๏Šฌ๏Šฉ๏Šฏ๏Šฎ๐‘ƒ+๐‘ƒ
  • C๏Šฉ๏Šฉ๏Šฎ๏Šฎ๐‘ƒร—๐‘ƒ
  • D๏Šจ๏Šฌ๏Šฉ๏Šฏ๏Šฎ๐‘ƒร—๐‘ƒ

Q11:

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

  • A๏Šฌ๏Šจ๐‘ƒ
  • B๏Šฎ๏Šจ๐‘ƒ
  • C๏Šฌ๏Šง๐‘ƒ
  • D๏Šฌ๏Šฌ๐‘ƒ

Q12:

Let ๐‘‹={๐‘ฅโˆถ๐‘ฅโˆˆโ„ค,โˆ’16โ‰ค๐‘ฅ<25} and ๐‘Œ={(๐‘Ž,๐‘,๐‘)โˆถ๐‘Ž,๐‘,๐‘โˆˆ๐‘‹, where ๐‘Ž,๐‘,๐‘}aredistinctelements. Which of the following represents the number of elements in ๐‘Œ?

  • A๏Šช๏Šง๏Šฉ๏Šฎ๐‘ƒ
  • B๏Šช๏Šจ๏Šฉ๏Šฏ๐‘ƒ
  • C๏Šช๏Šจ๏Šฉ๐‘ƒ
  • D๏Šช๏Šง๏Šฉ๐‘ƒ

Q13:

Find the solution set of 42๐‘ƒ=๐‘ƒ๏—๏Šฐ๏Šฉ๏Šฉ๏—๏Šฐ๏Šซ๏Šซ.

  • A{11}
  • B{4}
  • C{22}
  • D{2}

Q14:

Evaluate ๏Šง๏Šจ๏Šฉ๏Šฉ๐‘ƒ.

  • A123ร—122ร—3
  • B123ร—122ร—121
  • C123ร—3
  • D123ร—124ร—125

Q15:

If ๏—๏Šฑ๏˜๏Šจ๐‘ƒ=12 and ๏—๏Šฐ๏˜๏Šซ๐‘ƒ=6,720, 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 16โ‹…15โ‹…14.

Q18:

Calculate ๏Š๏Ž๏Š๏Šฑ๏Šง๏Ž๏Šฑ๏Šง๐‘ƒโˆถ๐‘ƒ.

  • A๐‘Ÿ๐‘›
  • B๐‘Ÿ
  • C๐‘›๐‘Ÿ
  • D๐‘›

Q19:

By using ๏Šจ๏Šฉ๏Ž๐‘ƒ=506 to find the value of ๐‘Ÿ, evaluate the expression ๏Šฉ๏Ž๏Šฐ๏Šฌ๏Šฉ๐‘ƒ.

Q20:

Calculate ๏Š๏Ž๏Š๏Ž๏Šฑ๏Šง๐‘ƒรท๐‘ƒ.

  • A๐‘›+๐‘Ÿ
  • B๐‘›โˆ’๐‘Ÿโˆ’1
  • C๐‘›โˆ’๐‘Ÿ
  • D๐‘›โˆ’๐‘Ÿ+1

Q21:

If ๏Š๏Šช๏Š๏Šฑ๏Šง๏Šฉ๐‘ƒ=7ร—๐‘ƒ, find ๏Š๏Šฐ๏Šฉ๏Š๏Šฑ๏Šฉ๐‘ƒ.

Q22:

If ๏—๏Šฑ๏˜๏Šช๐‘ƒ=73,440 and ๏—๏Šฐ๏˜๏Šง๐‘ƒ=26, find ๏—๏˜๐‘ƒ.

Q23:

Evaluate the expression ๏Š๏Šฐ๏Šช๏Š๏Šฑ๏Šช๐‘ƒ, given that ๏Š๏Šฐ๏Šซ๏Šญ๏Š๏Šฐ๏Šช๏Šฌ๐‘ƒ=9ร—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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