Examkin

Question

Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A university research department needs to form a 4-person delegation consisting of exactly 2 senior researchers and 2 junior researchers. The pool of available candidates contains 5 senior researchers and 6 junior researchers. However, 2 specific senior researchers refuse to serve on the delegation together. How many different 4-person delegations can be formed under these conditions?

  1. A
    120
  2. 135Answer
  3. C
    150
  4. D
    270
  5. E
    300

Answer

135 different 4-person delegations can be formed.
To calculate the number of valid delegations, first find the unrestricted number of delegations by multiplying the combinations of senior researchers (52)=10\binom{5}{2} = 10 and junior researchers (62)=15\binom{6}{2} = 15, which equals 150. Next, calculate the number of invalid delegations where both restricted senior researchers are chosen together, which is (22)×(62)=15\binom{2}{2} \times \binom{6}{2} = 15. Subtracting the invalid delegations from the total gives 150−15=135150 - 15 = 135.

Step-by-Step Solution

1
Calculate the total number of possible delegations without any restrictions.
(52)×(62)=10×15=150\binom{5}{2} \times \binom{6}{2} = 10 \times 15 = 150
Choosing 2 senior researchers out of 5 requires combinations, (52)=10\binom{5}{2} = 10. Choosing 2 junior researchers out of 6 requires combinations, (62)=15\binom{6}{2} = 15. By the Fundamental Counting Principle, the total number of unrestricted delegations is 10×15=15010 \times 15 = 150.
2
Calculate the number of invalid delegations that include both restricted senior researchers.
(22)×(62)=1×15=15\binom{2}{2} \times \binom{6}{2} = 1 \times 15 = 15
If both conflicting senior researchers are placed on the delegation, there is only (22)=1\binom{2}{2} = 1 way to select the senior members. The 2 junior researchers can still be selected in (62)=15\binom{6}{2} = 15 ways, giving 15 invalid delegations.
3
Subtract the number of invalid delegations from the total possible delegations.
150−15=135150 - 15 = 135
Subtracting the 15 restricted delegations from the 150 total delegations leaves 135 valid delegations.

Key Concept

Combinations with Exclusion Restrictions and Fundamental Counting Principle
Rate this question

Topics

Permutations, Combinations, and Fundamental Counting Principle

More questions from this topic