Worksheet: Properties of Combinations

In this worksheet, we will practice solving problems involving combinations.

Q1:

Find the value of ๐‘ฅ such that ๏Š๏Šช๐ถ=3,060 and ๏—๏Š๏—๏Š๏Šฐ๏Šช๐ถ=๐ถ.

Q2:

Find ๐‘Ÿ given that ๏Šง๏Šฆ๏Šซ๏Ž๐ถ>1 and ๏Ž๏Šง๏Šฆ๏Šฉ๐ถ>1.

Q3:

Find the value of ๏Šช๏Šฆ๏Šช๏Šง๏Šช๏Šจ๏Šช๏Šฉ๏Šช๏Šช๐ถโˆ’๐ถ+๐ถโˆ’๐ถ+๐ถ.

Q4:

If ๏Š๏Šฉ๐ถ=120, find ๐‘›.

Q5:

If ๏Š๏Šช๐ถ=35, find ๏Š๏Š๏Šฑ๏Šจ๐ถ.

Q6:

If (3๐‘›โˆ’5)!=24, find ๏Š๏Š๏Šฑ๏Šง๐ถ.

Q7:

Given that ๏Š๏Šฉ๐ถ=2๐‘›, evaluate ๏Š๏Šฉ๐ถ.

Q8:

Given that ๏Š๏Šช๐ถ=52๐‘›, determine ๏Š๏Šฉ๐ถ.

Q9:

Find the value of ๏Š๏Šซ๐ถ given that ๏Š๏Šฉ๐ถ=3013๐‘›.

Q10:

If ๏‰๏Šง๐ถ=2 and ๏‰๏Šฐ๏Š๏Šจ๐ถ=21, find (๐‘›โˆ’๐‘š)!.

Q11:

Which of the following inequalities is true given that ๏Š๏Šฉ๏Šง๏Š๏Šฎ๐ถ>๐ถ?

  • A๐‘›>39
  • B๐‘›=39
  • C๐‘›<39

Q12:

If ๐‘›!=5,040 and ๏Š๏Šฐ๏Šง๏Ž๏Šฑ๏Šจ๐ถ=70, find ๏Š๏Šฐ๏Šง๏Ž๏Š๏Šฐ๏Šง๏Ž๏Šฑ๏Šง๐ถ+๐ถ.

Q13:

Find the value of ๏Šซ๏Šฆ๏Šซ๏Šง๏Šซ๏Šจ๏Šซ๏Šซ๐ถ+๐ถ+๐ถ+โ‹ฏ+๐ถ.

Q14:

If ๏Šฏ๏Ž๏Šฏ๏Šฉ๐ถ=๐ถ, find all the possible values of ๐‘Ÿ.

  • A6
  • B3
  • C3 or 9
  • D3 or 6
  • E9

Q15:

If ๏Š๏Šฑ๏‰๏Šจ๐‘ƒ=5,402, and ๏Š๏Šฐ๏‰๏Šง๏Šฆ๏Šฉ๐ถ=104, find ๐‘š and ๐‘›.

  • A๐‘š=89, ๐‘›=15
  • B๐‘›=89, ๐‘š=15
  • C๐‘›=74, ๐‘š=178
  • D๐‘›=178, ๐‘š=74

Q16:

Given that ๏Š๏Ž๏Š๏Ž๏Šฐ๏Šจ๏Š๏Ž๏Šฐ๏Šฌ๐ถโˆถ๐ถโˆถ๐ถ=2โˆถ15โˆถ15, determine ๐‘› and ๐‘Ÿ.

  • A๐‘›=12, ๐‘Ÿ=14
  • B๐‘›=12, ๐‘Ÿ=3
  • C๐‘›=12, ๐‘Ÿ=2
  • D๐‘›=13, ๐‘Ÿ=2
  • E๐‘›=15, ๐‘Ÿ=2

Q17:

Given that ๏Š๏Šฐ๏Šง๏Ž๏Šฐ๏Šง๏Š๏Šฐ๏Šง๏Š๏Šฑ๏Ž๐‘ƒโˆถ๐ถ=20! and ๏Š๏Šฐ๏Šง๏Ž๏Š๏Šฐ๏Šง๏Š๏Šฑ๏Ž๏Šฐ๏Šจ๐ถโˆถ๐ถ=5โˆถ19, find ๐‘› and ๐‘Ÿ.

  • A๐‘›=22, ๐‘Ÿ=19
  • B๐‘›=22, ๐‘Ÿ=5
  • C๐‘›=23, ๐‘Ÿ=19
  • D๐‘›=25, ๐‘Ÿ=22

Q18:

Which of the following is equal to ๏Šช๏Šง๏Šซ๐ถ?

  • A๏Šช๏Šง๏Šซ๐‘ƒร—5!
  • B๏Šช๏Šง๏Šซ๐‘ƒร—5
  • C๏Šช๏Šง๏Šซ๐‘ƒ5!
  • D๏Šช๏Šง๏Šซ๐‘ƒ5

Q19:

Determine the value of ๏Šจ๏Šฉ๏Šฎ๏Šจ๏Šฉ๏Šฌ๐ถ๐ถ without using a calculator.

  • A734
  • B43
  • C73
  • D52
  • E347

Q20:

Find ๐‘ฅ and ๐‘ฆ, such that ๏—๏˜๏—๏˜๏Šฐ๏Šง๐ถ=38ร—๐ถ and ๏—๏˜๏—๏˜๏Šฐ๏Šฌ๐ถ=๐ถ.

  • A๐‘ฅ=9, ๐‘ฆ=2
  • B๐‘ฅ=4, ๐‘ฆ=2
  • C๐‘ฅ=10, ๐‘ฆ=2
  • D๐‘ฅ=12, ๐‘ฆ=4
  • E๐‘ฅ=2, ๐‘ฆ=2

Q21:

If ๐‘›!=๐ถร—๐‘ƒ๏Šฏ๏Šฉ๏Šซ๏Šฉ, find ๐‘›.

Q22:

If ๏Š๏Ž๏Š๏Ž๐‘ƒ=๐ถ, find the value(s) of ๐‘Ÿ.

  • A3 or 4
  • B3
  • C4
  • D2
  • E0 or 1

Q23:

Find the value of ๏Š๏Šซ๏Ž๏Šฐ๏Šจ๐ถ, given that ๏Š๏Ž๏Šฐ๏Šช๏Ž๏Š๏Šช๏Ž๏Šฑ๏Šซ๐ถ=๐ถ๏Žก and ๏Š๏Šง๏Šจ๐ถ=455.

Q24:

Find ๏Š๏Ž๏Š๏Ž๐‘ƒ๐ถ.

  • A1
  • B๐‘›!
  • C(๐‘›โˆ’๐‘Ÿ)!
  • D๐‘Ÿ!
  • E(๐‘›โˆ’1)!

Q25:

Using the equations ๏Š๏Ž๏Šฐ๏Šฉ๏Š๏Ž๏Šฐ๏Šจ๐ถโˆถ๐ถ=34โˆถ13 and ๏Š๏Šฐ๏Šง๏Ž๏Š๏Ž๐‘ƒโˆถ๐‘ƒ=47โˆถ37 to find the values of ๐‘› and ๐‘Ÿ, evaluate the expression ๐‘›+๐‘Ÿ.

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