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Zorluk: KolayPermutations, Combinations, and Fundamental Counting Principle

A committee is to be selected from a group of 55 distinct people: PP, QQ, RR, SS, and TT. Which of the following statements regarding the possible selections or arrangements of people from this group are true? Select all that apply.

  1. The number of different 22-person committees that can be formed from the group is 1010.Cevap
  2. The number of different 33-person committees that can be formed from the group is 1010.Cevap
  3. C
    The number of different ways to assign 22 distinct officer roles (President and Vice President) from the group is 1010.
  4. D
    The total number of different ways to arrange all 55 people in a single line is 2525.
  5. The number of different ways to select and arrange 33 of the 55 people in a line is 6060.Cevap

Cevap

The statements confirming that 1010 different 22-person committees can be formed, 1010 different 33-person committees can be formed, and 6060 different 33-person linear arrangements can be formed are all correct.
Selecting committees without specific roles requires combinations (nk)\binom{n}{k}, giving (52)=10\binom{5}{2} = 10 and (53)=10\binom{5}{3} = 10. Arranging 3 people in ordered positions requires permutations P(5,3)=5×4×3=60P(5,3) = 5 \times 4 \times 3 = 60. Thus, all three corresponding statements are correct.

Adım Adım Çözüm

1
Evaluate the 2-person committee selection statement.
\binom{5}{2} = \frac{5 \times 4}{2} = 10
Selection of a committee without specific roles is an unordered combination.
2
Evaluate the 3-person committee selection statement.
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
Choosing 3 items out of 5 yields the same number of outcomes as choosing 2 items out of 5.
3
Evaluate the 2-person officer assignment statement.
P(5,2) = 5 \times 4 = 20
Assigning distinct officer positions means order matters, requiring permutations rather than combinations.
4
Evaluate the 5-person line arrangement statement.
5! = 120
The total number of linear arrangements of 5 distinct objects is given by 5 factorial.
5
Evaluate the 3-person line arrangement statement.
P(5,3) = 5 \times 4 \times 3 = 60
Ordering 3 out of 5 people in a line uses the fundamental counting principle with decreasing choices per slot.

Anahtar Kavram

Distinguishing between combinations (where selection order does not matter) and permutations (where selection or position order does matter).
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Konular

Permutations, Combinations, and Fundamental Counting Principle

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