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Permutations, Combinations, and Fundamental Counting Principle

22 soru

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Soru 21Soru →

A technology committee has a pool of 88 available guest speakers consisting of 55 computer scientists and 33 data privacy experts. A 44-person panel is to be formed from this pool. Which of the following statements regarding the possible panel selections must be true? Select all such statements.

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Cevap: The total number of different 44-person panels that can be formed without any restriction on specialty is equal to 7070.; The number of different 44-person panels that consist of exactly 22 computer scientists and 22 data privacy experts is equal to 3030.; The number of different 44-person panels containing at least 11 data privacy expert is equal to 6565.

Cevap

The correct statements are those asserting that total unrestricted panels equal 70, panels with exactly 2 computer scientists and 2 privacy experts equal 30, and panels with at least 1 privacy expert equal 65.
The total unrestricted 4-person panels from 8 speakers is 8C4 = 70. Selecting 2 computer scientists (5C2 = 10) and 2 privacy experts (3C2 = 3) yields 10 × 3 = 30 panels. Using complementary counting, panels with at least 1 privacy expert equal total panels (70) minus panels composed entirely of computer scientists (5C4 = 5), giving 70 - 5 = 65.

Adım Adım Çözüm

1
Calculate unrestricted combinations of 4 speakers out of 8
(84)=8×7×6×54×3×2×1=70\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
Selection order does not matter for committee membership, so we use combinations.
2
Calculate combinations with restricted counts (2 CS and 2 PE)
(52)×(32)=10×3=30\binom{5}{2} \times \binom{3}{2} = 10 \times 3 = 30
By the Fundamental Counting Principle, independent choices for each subgroup are multiplied.
3
Calculate combinations with 'at least 1' condition using the complementary counting method
Total panels minus panels with 0 PE: 70−(54)=70−5=6570 - \binom{5}{4} = 70 - 5 = 65
Subtracting outcomes that violate the constraint from total possible outcomes simplifies 'at least 1' calculations.
4
Evaluate the permutation and impossible constraint statements
Arranging 4 CS out of 5 gives P(5,4)=120≠20P(5,4) = 120 \neq 20. A 4-person panel with 0 CS requires 4 PE out of 3, which gives 0≠10 \neq 1.
Arrangement requires permutations, and panel requirements exceeding available pool members yield zero valid groups.

Anahtar Kavram

Combinations, Permutations, and Complementary Counting Principle
Soru 22Soru →

A librarian is arranging 77 distinct books on a display shelf: 44 history books and 33 science books. If all 33 science books must be placed adjacent to one another, in how many different linear orders can the 77 books be arranged?

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Cevap: 720

Cevap

720 linear arrangements
To arrange items with an adjacency restriction, bundle the restricted items (the 3 science books) into 1 block. This leaves 4 history books and 1 science block, totaling 5 units to arrange linearly in 5! = 120 ways. Within the science block, the 3 distinct books can be arranged in 3! = 6 ways. By the Fundamental Counting Principle, the total number of linear arrangements is 5! × 3! = 120 × 6 = 720.

Adım Adım Çözüm

1
Group the constrained items into a single block.
Treat the 33 distinct science books as a single unit or block, denoted as [S][S].
Since the science books must always stand next to each other, they move as a single combined item.
2
Count the total number of items to arrange linearly.
We have 44 individual history books plus 11 science block, giving 4+1=54 + 1 = 5 items.
The 55 units can be ordered among themselves in 5!5! ways.
3
Calculate the outer arrangements of the 5 units.
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
This accounts for all position permutations of the history books and the block.
4
Calculate internal permutations within the science block and multiply.
3!=63! = 6 ways within the block. Total arrangements = 120×6=720120 \times 6 = 720.
By the Fundamental Counting Principle, total ways equals outer permutations multiplied by inner permutations.

Anahtar Kavram

Permutations with Adjacency Restrictions (Block Method)
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Permutations, Combinations, and Fundamental Counting Principle Alıştırma Soruları — GRE General Test — Sayfa 2 | Examkin