Examkin

Question

Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

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.

  1. The total number of different 44-person panels that can be formed without any restriction on specialty is equal to 7070.Answer
  2. The number of different 44-person panels that consist of exactly 22 computer scientists and 22 data privacy experts is equal to 3030.Answer
  3. The number of different 44-person panels containing at least 11 data privacy expert is equal to 6565.Answer
  4. D
    The number of ways to select and arrange 44 computer scientists in a specific speaking order on stage is equal to 2020.
  5. E
    The number of different 44-person panels containing no computer scientists is equal to 11.

Answer

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.

Step-by-Step Solution

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.

Key Concept

Combinations, Permutations, and Complementary Counting Principle
Rate this question

Topics

Permutations, Combinations, and Fundamental Counting Principle

More questions from this topic